Achromatic devices with thermal radiation sources

ABSTRACT

A light emitting assembly comprising at least one of each of a solid state device and a thermal radiation source, couplable with a power supply constructed and arranged to power the solid state device and the thermal radiation source, to emit from the solid state device a first, relatively shorter wavelength radiation, and to emit from the thermal radiation source non-visible infrared radiation, and a down-converting luminophoric medium arranged in receiving relationship to said first, relatively shorter wavelength radiation, and the infrared radiation, and which in exposure to said first, relatively shorter wavelength radiation, and infrared radiation, is excited to responsively emit second, relatively longer wavelength radiation. In a specific embodiment, monochromatic blue or UV light output from a light-emitting diode is down-converted to white light by packaging the diode and the thermal radiation device with fluorescent or phosphorescent organic and/or inorganic fluorescers and phosphors in an enclosure.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is a divisional of U.S. Nonprovisional application Ser.No. 16/699,788 filed 2 Dec. 2019 (12/02/2019) which is incorporatedherein in its entirety.

TECHNICAL FIELD

This invention relates to solid state light emitting devices such aslight emitting diodes and more particularly to such devices whichproduce white light.

BACKGROUND OF THE INVENTION

Devices based on converting radiation energy into electrical or chemicalenergy suffer when the spectrum of the incident radiation does notoverlap the spectral absorptivity of the absorbing species. When thereis perfect overlap, all incident radiation can theoretically be absorbedand can be used: we call this the resonance effect. As an example ofresonance, consider in electron spin resonance (ESR), an unpairedelectron can move between the two energy levels by either absorbing oremitting a photon of energy hv, such that the resonance condition,hv=ΔE, is obeyed. When there is perfect resonance between the appliedmagnetic field and the incident radiation, a strong signal is obtainedreflecting a strong transition.

When there is less than perfect spectral overlap, frequently heat isgenerated, the temperature rises, and passive or active cooling may berequired. Recent attempts at improving photovoltaic panels, and takingadvantage of the resonance effect, use vertically stacked cells, suchthat each layer—using different materials in each layer—is designed tocapture cumulatively and sequentially all of the wavelengths of solarradiation incident on the panel. In the absence of resonance, solarradiation has been used for years for rooftop heating of water,ultimately for inside use. Perhaps except for purposeful heating, suchas for indoor water consumption. it is generally the case that theproduction of heat is unwanted in most photo-devices.

For the use of light-emitting diodes, the input is electrical, and theoutput is radiation: it operates in a manner opposite that of aphotovoltaic device. Core to the instant invention is to manage andmanipulate the entropy of radiation and its proclivity for notconserving photon numbers, normally considered only applicable toemission of thermal radiation, but even with luminescent emission, it isobserved. (Shishkov, 2017) Parenthetically, all matter generates andemits electromagnetic waves (i.e.; radiation, or light) based on thetemperature of the matter.

For light-emitting diodes, the efficiency of converting electricalenergy into photonic energy can be accomplished—depending on thematerials used and their bandgaps—by either impacting 1) the averageenergy of each electro-magnetic wave (the exiting photonic energy) and,or 2) the number of modes of each wave. Using the terminology ofparticles, by impacting 1) the average energy of all photons and, or 2)the number of photons.

Chemical Potential is a term that is derived from thermodynamics. It andentropy are the stars of the instant invention. Chemical Potential isthe free energy added to a system with the addition of each particle;for example, Gibbs free energy per number. A concise introduction tothermodynamics is beyond the scope of this disclosure but one summarizedby others is provided as a short introduction:

“The first law of thermodynamics states that energy can neither becreated nor destroyed—it can only change in form. However, the secondlaw is the most discussed. Although difficult to define uniquely, itgenerally relates to the spontaneity of processes in systems not atequilibrium. In 1865, Rudolph Clausius first formulated this law inrelation to the flow of heat, which cannot occur spontaneously from acold body to a hot body. Since, it has been reformulated in terms ofentropy, which is accepted as increasing in the universe over time.Entropy producing processes are spontaneous, but the rate at which theyoccur is not specified. The third law states that the entropy of asystem will approach a minimum value as that system approaches absolutezero temperature. From the scientific point of view, the universe solelycomprises matter and energy, two items declared to be interchangeable byEinstein. All systems are thus complexes of energy and matter, whosedelineation depends on an observer, with arbitrary boundaries open totransfers of energy and matter. We point out that natural systems arecharacterized by nested scales of action, with that of smaller systemsexisting inside a hierarchy of larger systems, the properties of eachscale contributing to the function of larger systems. We can define aclosed system as one with no matter transfer across its boundaries, andan isolated system with no transfer of either energy or matter. However,both these definitions can only be considered as approximations. Infact, all ecosystems are open, the earthly systems essentiallycomprising living systems of the biosphere, acting in concert with thethree abiotic systems, the lithosphere, the hydrosphere, and theatmosphere (representing solids, liquids, and gases) comprising overallthe ecosphere.” (Rose, Crosssan, & Kennedy, 2008) It is emphasized thatentropy is applicable for spontaneous processes and those that are notat equilibrium. These are referred to as irreversible processes.

The thermodynamic Detailed Balance is used to increase photon numbersand thereby to increase the efficiency of light emitting diode lampswithout the need for cooling. The system uses gas with reasonablethermal diffusivity or thermal effusivity to transfer heat produced atthermal sources and or sources of primary radiation to sources ofsecondary radiation. At the source of secondary radiation, gas is usedto retain the heat at this source and to protect or enhance the sourceand its emission yields, respectively.

The principle of Detailed Balance means that for a process atequilibrium and observed on a macroscopic level, the forward rate ofeach step (absorption of radiation) is equal to the reverse rate of thatstep (emission of radiation). We are concerned with the Detailed Balanceat the source of secondary radiation. A thermodynamic system atequilibrium, is one where it is not changing with time and thetemperature, and concentration, are uniform throughout the system,unless otherwise perturbed and in which reaction to said perturbation, anew equilibrium is obtained. Thus, one can look at Detailed Balance fromthe perspective of kinetics to reach equilibriums. A fluctuationtemporarily disturbs the system which returns back to equilibrium onceover.

A system that is at a steady state is one in which one of theconcentrations is not substantively changing and thus an approximationcan be applied to the kinetics of the system. (Alberty, 2004) For theinstant invention, thermodynamic equilibrium, local equilibriumassociated from fluctuations from equilibrium, non-equilibriumthermodynamics, and also kinetics, are important to the instantinvention. The instant invention is directed towards GeneralIllumination.

Notwithstanding the above, our principle focus is on Detailed Balancefrom the perspective of entropy for a photophysical process thatconverts incident radiation (the primary radiation) to exiting radiation(the secondary radiation from the phosphor) such that the change inentropy for the process is minimal. With each new photon added to aradiation field, there is an increase in entropy; with each exitingphoton, there is a decrease in entropy from the radiation field. Impliedby this consideration is that a Detailed Balance mismatch betweenincident radiation entropy and exiting radiation entropy signifies aninefficiency of a steady state photophysical process employed forGeneral Illumination. This is usually expressed in the manner that theenergy of the incident Primary Radiation which is absorbed is muchgreater than the energy of the exiting secondary radiation which isemitted, the difference to which is transmitted to the surroundings inthe form of entropy such that there is a total (system and surrounding)balance in entropy at equilibrium. For a steady state the overallentropic process leads to an increase in entropy for each GeneralIllumination cycle, but the rate of entropy increase does not changeover time. Quantum Yields are based on actinometry.

Modern implementations of General Illumination has been an importantindustry since before the Industrial Revolution offering greatemployment and essential innovation for over, at least, one century,inuring to the benefit of society who can enjoy the pursuit of work andleisure even when the sun is down. (Bright, 1949) The importance of thiscommercial pursuit is unlikely to change even with the now profitablemigration towards achromatic, near-perpetual solid-state lighting(ANSL), near-perpetual in the sense of the devices long-term operationalstability, in term of decades, as opposed to the built-in obsolescenceof incandescent lamps.

More specifically and presently, prior to the instant invention, ANSLlamps constructed with light emitting diodes (LEDs) for GeneralIllumination converts a portion of primary radiation, of a certainenergy and entropy, emitted by the diodes, to secondary radiation, of adifferent energy and entropy, emitted by the phosphors, which, whencombined with reflected (scattered) primary radiation, appears to thehuman observer as white light illuminance (as well as luminance) andallows for the process of vision (conversion of energy into useful anddesired work). In the most typical implementation, prior to the instantinvention and among several others, the primary radiation is observed byhumans as blue and the secondary radiation, absent combination with thereflected (scattered) blue radiation, is observed as yellow. Thiswide-ranging process used for General Illumination is calledDown-Conversion and it is used to prepare LED-based ANSL as well as, ifso desired and suitably constructed, chromatic analogues. The instantinvention calls the emission from the phosphor secondary emission orsecondary radiation and which may be fluorescence. [We capitalizeDown-Conversion in the instant invention as it is the name of a uniqueprocess hereinafter defined and to highlight the meaning of same whenused in the disclosure. For the same reason we herein capitalize theverb Down-Converting and the nouns General Illumination and QuantumYields.]

This Down-Conversion approach puts a commercial premium on theavailability of LED dies that appear, absent any radiation mixingprocess, blue. It is for this reason that the development of blue LEDdies, otherwise of little technical nor societal value, enjoyedsignificant commercial success (because of its utility when perfected)and scientific notice (because of the long-term difficulty to, in fact,perfect). A similarly related lighting technology can be designed usingLED dies higher in primary radiative energy than that which appears tothe observer as blue (e.g.; violet or ultra-violet).

The primary radiation that originates from LEDs is electro-initiatedluminescence (i.e.; electroluminescence) meaning that an electricalcurrent (ampere) of a suitable force (volts) is used to populate excitedstates of the corresponding energy (electron-volts or Joules); thesestates subsequently and spontaneously produce primary radiation (anelectromagnetic field) of the same or different magnitude in energy(expressed in frequency, or wavelength, or wavenumbers) compared withthe energy for initial population of the relevant electronic states. Thesecondary radiation so produced is photo-initiated luminescence (i.e.;photoluminescence); the body that luminesces is called a luminophore andmore often a phosphor (without regard to any changes in spinmultiplicity of the transition).

Luminescent emission is spontaneous, and, under certain but limitedconditions, stimulated emission predominates. Consequently, ANSLcontains two forms of radiation mixed together (electro-luminescence andphotoluminescence); further the combination is from one that is absorbedand one that is scattered. Note that absorbed radiation, if fullyabsorbed as in a blackbody, has an absorptivity of one (1) and scatteredradiation, if fully scattered such as in a white body, has anabsorptivity of zero (0).

Radiation's interaction with matter, wherever it may be and however itmight be studied, is essentially one of absorption, scattering andtransport. The Feynman model for refractive index, ingenuous that it is,but which we shall not further explain, is an example of radiationinteraction with matter for the purpose of transport. We consider theedge of the ANSL device to be the environmental interface through whichenergy escapes to perform work elsewhere.

The photo-initiated process in Down-Conversion is called trivial energytransfer which also transfers through space, without the assistance ofintervening matter, entropy; other means of energy transfer, such asFörster Resonance Energy, to effect subsequent luminescence, areavailable. We note that energy transfer generally and without furtherspecification, and more importantly, entropy transfer, is essentiallythe instant invention's focus in pursuit of these arts: GeneralIllumination. To the knowledge of this inventor, the entropy of FörsterResonance has heretofore not been reported, however since a specificorientation is required, no doubt the process involves a decrease inentropy to obtain the orientation, requiring an increase in entropysomewhere else. One normally thinks of Förster Resonance Energy Transferas being isoelectronic (of the same energy). Traditionally, FörsterResonance Energy Transfer has not been used as a quantitativealternative to trivial energy transfer, from the excited state of theprimary emitter to populate the secondary emitter, as used inDown-Conversion, to make ANSL. (Itskos, Othonos, Choulis, & Illiopoulos,2015)

Thermodynamics of engineering devices, such as heat engines, is an olddiscipline, that remains a powerful and incontrovertible science, fromwhich the energy output and energy efficiency of engines cannot escape,even today. Subsequently, the study of thermodynamics to chemicalprocesses further elucidated this science to reveal its universalapplicability (as brilliantly discussed by William B. Jensen in “GeorgeDowning Liveing and the Early History of Chemical Thermodynamics”)beyond mechanical devices. (Jensen, 2013) While thermodynamics cannotexplain all chemical and physical phenomena, which resulted in theperhaps reluctant development of quantum mechanics and its powerfulelucidation when applied to chemical—quantum chemical (QC)—processes,it, thermodynamics, yielded initial evidence of conservation laws ofnature that cannot be violated. This was and remains its very essence.Just as the expansion of thermodynamics into the chemical sciences wasinstrumental in expanding our understanding of its macroscopic relevanceto efficiencies, conservation and probabilities, the same expansion ofquantum mechanics to quantum chemistry played a fundamental role in ourunderstanding microscopic and atomic (or molecular or solid-state)possibilities.

Photochemical processes, those initiated by radiation, regardless of thesource and the radiation's characteristics, have been well studied overthe last century with a particular focus on quantum mechanical selectionrules and observed spectroscopy for electronic transitions betweenpossible states as well as a plethora of kinetic examinations due to thefact that these excited state processes occur in time spans relativelyshort compared to ground state endothermic reactions. Whereas the studyof the thermodynamics of photochemical processes is not as oftenconsidered with the quantum mechanical discovery of transition selectionrules, it is ironic that quantum mechanics itself evolved from thethermodynamic study of photonic radiation, in particular that classcalled thermal radiation.

Whereas photosynthesis on the planet earth, fueled by thermal radiationfrom the sun, has long been studied from a thermodynamic perspective,more recently the study of photovoltaic devices—solar cells—hasreconsidered the limits of efficiency from a thermodynamic perspectiveeven as its quantum chemical study remains an important component ofdevice design and process understanding and manipulation. Of course,like photosynthesis, the incident radiation that initiates thephotovoltaic process in solar cells is of thermal radiation, from thesun, although photovoltaic devices are obviously of commercial benefiteven with radiation from unnatural sources. (Ye, et al., 2017) Both withphotosynthesis and photovoltaics and solar cells, the source of primaryradiation is external to the device and the same source carries with itthe curious benefits of maximal thermal radiation, and the entropyassociated therewith, subsequent to which energy conversion or energytransfers, as well as entropy transfer, to matter, takes place. Theentropy of thermal radiation has been studied for a long period of time,as we shall hereinafter remind, and the efficiency of energy conversionand energy transfer from thermal radiation-initiated processes, nodoubt, both are influenced by thermodynamic entropy.

In our initial considerations, we link energy with entropy andacknowledge that there is a mismatch in the energy emitted versus theenergy absorbed at the source of secondary radiation, the phosphor. Ithas been assumed, as we shall hereinafter comment, that this reductionin energy at the phosphor results in an increase in heat at thephosphor. However, General Illumination is a function not of energy, butradiometric power conveyed in the colloquially context of intensity. Inthe field of radiometric thermodynamics, however, one's principal focusis on energy density and entropy density as they are influenced byunderlying intensities necessary to offer General Illumination atpractical and practicable levels.

The current interest in the utilization of thermal radiation (Greffet,Bouchon, Brucoli, & Marquier, 2018) in engineered devices (Tsakmakidis,Boyd, Yablonovitch, & Zhang, 2016) emanates from the fact that theprimary radiation is economically available: from the Sun. Howeverthermal radiation is also associated with heat, which we cannotoverlook, and that is recognized as a source of energy inefficiencies,that may be initiated by and or spontaneously dissipated by radiation(as well as other means). Accompanying, thermal radiation has somepeculiar properties different from other forms of radiation that maybeuseful in the manipulation of as an incident beam and to maximize energyefficiency and to overcome the conventional theoretical limitsheretofore observed for photovoltaic or other optoelectronic devices.(Soares, Ferro, Costa, & Monteiro, 2015) For example, as describedelsewhere, “tuning the color of the blackbody radiation in nanoparticlesby harvesting the low energy photons into the visible spectral regionwas found to be possible by adjusting the excitation power, paving theway to further developments of these nanoparticles for lighting andphotovoltaic applications”. (Soares, Ferro, Costa, & Monteiro, 2015) Atthis stage, unfortunately, this referenced device was operated between12 K and ambient, not a viable implementation for commercial purposes.It is asked, however, that one notice the focus on power as theinfluencing step.

By way of another example, among the interesting applications is the useof nanoantennae to dramatically enhance the rate of spontaneous emission(or dramatic reduction in radiative lifetime) from organic luminescentdyes by several orders of magnitude. This is a rather herculeanachievement, (Tsakmakidis, Boyd, Yablonovitch, & Zhang, 2016) in thatthere is no change in spin multiplicity causing this effect (i.e.;changes in a processes' spin multiplicity, a quantum mechanical process,are known to dramatically change radiative lifetimes).

Hereinbefore, luminescence has been defined in the context of generatingsecondary radiation (often fluorescence) after an electronic transitionto a higher electronic state populated as a consequence of absorption ofincident primary radiation. Thermal radiation, on the other hand, hasbeen loosely defined as originating from the Sun or as a means ofdissipating heat from a heat source, like the Sun.

Nevertheless, thermal radiation and luminescence are intricately linkedalthough they only appear to be somewhat similar if not completelydifferent phenomena, and both are observed concurrently in suitablesystems. Following Greffet et. al.: (Greffet, Bouchon, Brucoli, &Marquier, 2018)

“We now turn to the modelling of electroluminescence andphotoluminescence using the local form of Kirchhoff law. While the usualpicture associated to these processes involves the radiativerecombination of an electron-hole pair, it is important to recognizethat thermal emission at the same frequency is also related to the samemicroscopic process. The difference between thermal radiation, on onehand, and electroluminescence and photoluminescence, on the other hand,stems from the process promoting electrons to the conduction band. Afterthermalization, the band states are occupied according to a Fermi-Diracdistribution both in the conduction and in the valence band. Yet, eachband has its own chemical potential called quasi-Fermi level to accountfor the modification of carrier density. This is possible because thethermalization takes place in typically 1 ps whereas radiativeelectron-hole recombination takes place in typically 1 ns. These twoquasi-Fermi level situations is the typical regime ofelectroluminescence for light-emitting diodes. Knowing the occupation ofthe conduction and valence bands, it is possible to compute therecombination processes and therefore to derive the emitted power. Thisprocedure has been outlined in detail by Wurfel [P. Wurfel, The ChemicalPotential of Radiation, J. Phys. C15, 3967 (1982)] for a homogeneousmedium. He introduced a generalized form of Kirchhoff law to modelelectroluminescence by a slab of semiconductor.” (Greffet, Bouchon,Brucoli, & Marquier, 2018)

This approach is also used when dealing with dye molecules where twobands can also be defined with fast internal relaxation processes andslower radiative recombination. This case is called the photo-molecularview; that with photoexcitation of semiconductors (as opposed toelectro-excitation) is often called the photoelectric view.

Before moving on, it is important to share with further elaboration thespecifics alluded to in the insightful treatment of Greffet et. al.which we call Greffet's rule. Therein it reminds that the two terms ofemission are luminescence and thermal radiation but that these twoprocesses of emission, for particular frequency, come from the samepopulated final electronic excited states just prior to the emissionsthemselves. The only difference in these emissions emanate from how thefinal electronic excited states came about to be populated in the firstplace: as at ambient and or without incident radiation, only aninexplicably minor amount of matter will find itself in the finalelectronic excited states from which emission takes place. Thedifference is that luminescence is preceded by excitation from theground state under the influence of electrical current of appropriateenergy (electroluminescence) or a radiation flux of suitable energy(photoluminescence). The alternative route is through heat ofappropriate temperature and the heat flux, in other words, energydensity, associated therewith.

Since Down-Conversion in ANSL concerns itself principally from theprocess of photoluminescence yielding the secondary radiation emissionusually fluorescence, we shall not further too-often dwell on theelectroluminescence that generates the primary radiation (which issubsequently incident upon matter or scattered by matter) and is thesource of heat due to the inefficiencies inherent inelectroluminescence. The instant invention is based in at least one parton the extraordinary concept that Greffet's rule applies not only to theGaN light-emitting diode in a LED lamp but also to the phosphor normallywith a Stokes shift in Down-Conversion. That is to say, the differencebetween luminescence and thermal radiation of spectral peaks isassociated with the route of “promoting electrons” either to the valenceband (in a semiconductor device) or to the higher molecular orbital (ina molecular device), subsequent to which the resulting emission stemsfrom the depopulation of the same electrons back to the ambient statefrom which it was initially populated. In other words, the route ofpopulation matters, and it seems (in a peculiar manner) to be recalledas the subsequent emission takes place, the entropy of the exitingradiation carries with it the information on the route of excitation.The information may not be easily decipherable, but it is no doubtencoded within the final radiation itself as it crosses the boundaryfrom the system to the surroundings. This may be a controversialconclusion for those that argue once a photon is absorbed its sourcecannot be discerned; nevertheless, entropy transfer to matter fromradiation forms the cornerstone of quantum mechanics and the DetailedBalance.

We shall hereinafter return to this means of recollection which,depending on one's point of view, may reveal that an incident source ofexcitation may have some control over the routes of de-excitation suchthat not all sources of excitation are the same and how the excitedstates are generated does indeed matter. To some extent, thisargumentation of the instant invention is reflective of the differenteffects of chiral radiation on achiral matter, so we would not posturethat the above discernment is totally revolutionary and withoutprecedence. On the other hand, the role of radiation with high entropyversus radiation with low entropy is rarely considered, even by experts,as having different fundamental outcomes, especially in the context ofDown-Conversion for ANSL. What we can say, however, is that thermalradiation emission has fundamentally different properties thanluminescence; more specifically, in the case of the former, the obviousindifference to the conservation of numbers whereas in the case of thelatter, the almost strict adherence to the conservation of numbers,except for very special unique cases which still complies with the lawof conservation of energy (e.g.; singlet fission of pentacene).

For the instant invention, however, we focus on that there may be avariety of ways to simultaneously populate the desired electronicexcited states of a phosphor, that upon depopulation yield the desiredemission frequencies and that these ways may be realized simultaneously,at least in a thermodynamic timeframe.

This thermodynamic understanding of photochemical processes, asdiscussed in the few preceding paragraphs, was noted, many years ago, byRoss whose words also bring important insight and at the time of itswritings uniquely perceptive, warranting wide-spread examination andteachings, and are of paramount importance to the instant invention:

“There are two ways of viewing the excitations caused by the absorptionof light: Excitations may be considered as producing an increase in thepopulation of electrons in a set of states of fixed number, with acorresponding decrease in the population of electrons in another set ofstates of fixed number.

“Alternatively, the identical process may be viewed as an increase inthe number of an excited state molecular species and a concomitantdecrease in the number of a ground state molecular species. These twopoints of view may be called the photoelectric and the molecularphotochemical views, respectively, and the distinction between the twobecomes meaningful when one considers' the coupling between thelight-absorbing molecules, and the larger system of which they are apart. If this system operates through electron migration, then thephotoelectric view is the more natural, while if the resultlight-absorption is molecular rearrangement, then the molecularphotochemical view may prevail.

“It is important to remember that the distinction is one of point ofview only, as influenced by the fate of the excitation, and makes nodifference in the thermodynamics of the light-absorption. The formalismfor the photoelectric case has been extensively developed for use insemiconductor systems. In the presence of complete thermal equilibrium,the chemical potential of all of the electrons is equal to the Fermilevel, which usually lies midway in energy between the ground band andthe excited state band. The electrons within the E band and within the Gband each have a chemical potential which is similarly defined, andwhich is called a quasi-Fermi level. As electrons are transferred from Gto E under the influence of light, the quasi-Fermi level of G dropsbelow the equilibrium Fermi level, and the quasi-Fermi level of Eincreases correspondingly.

“The alternative point of view is to consider changes in the partialmolar free energy of the light-absorbing molecules in their ground stateand in their excited state. The action of light usually depletes thepopulation of the ground state molecules only very slightly, alteringthe chemical activity of these species to a negligible extent; in thiscase the potential difference arising between the bands is due almostentirely to the greatly increased population of molecules in the excitedstate.” (Ross R. T., 1967) We note that Ross' statements were made in1967, more than a decade before the incorporation of chemical potentialinto a generalized Planck's formula (Planck's equation) referencedhereinbefore due to Würfel.

The important comment above distinguishes the processes based on theprocess for promoting (exciting) electrons, and not the subsequentspontaneous depopulation: in the case of a photovoltaic device,depopulation by creating an electric field; in the molecular device, bycreating a radiation (electromagnetic) field. Of course, at normalambient temperatures, the population of those excited states by thermalpromotion and emitting at the frequency of interest is minimal. This iseven true for a perfect black body or for matter simulating same in aless than perfectly reflecting isothermal cavity. We can see in Ross,the emergence of a statistical mechanics view of entropy describing thestates of the light-absorbing matter. (Ross R. T., 1967)

The process of Down-Conversion for General Illumination of ANSLessentially converts some primary radiation of a frequency range υ₁ to amore useful frequency range υ₂. As noted by Greffet, et. al.,electro-luminescence (the opposite of photovoltaic or photoelectric) orphotoluminescence, on one hand, and thermal radiation at the samefrequency, on the other, is related to the very same microscopicprocess. Nevertheless, there is little discussion of the thermodynamicsof traditional photoluminescence when the incident radiation is notthermal radiation; on Earth, when the incident radiation is not from theSun. In the alternative, the study of photosynthesis on Earth, poweredby the Sun, and photovoltaic solar cells, also powered by the Sun, havelong benefited from, even if only intermittently, from thermodynamicconsiderations, which famously is reported to be rediscovered again andagain.

Interestingly, with the post-Industrial Revolution's impact in globalwarming, reaching an extremum not hereinbefore anticipated, climaticstudies of the Earth are also beneficiaries of thermodynamic analyses,which we shall briefly comment so as to show our motivation for theinstant invention.

Entropy, as a property of state (pressure, temperature, and volume) andirrespective of the path undertaken by a system of molecules to arriveat that condition, is therefore an important feature of atmosphericgases. Indeed, its capacity to explain how thermal radiation may beabsorbed and partitioned into different degrees of freedom of motion mayexplain the warming potential of greenhouse gases. (Kennedy, Geering,Rose, & Crossan, 2019) Action, a physical property, is related to thevector and angular momentum, with similar dimensions of mass by velocityby inertial radius [mvr], but is a distinct scalar quantity independentof direction. Like entropy, it is an extensive or cumulative property,but with physical dimensions of the integral of energy with time, or ofthe instantaneous angular momentum with respect to angular motion;classically, Action is considered as the integral of momentum withdistance. (Kennedy, Geering, Rose, & Crossan, 2019)

As a variable property of conservative systems, Action has beenconsidered to take stationary values, a result sometimes referred to asthe principle of least action, mentioned hereinbefore in this instantinvention. (Kennedy, Geering, Rose, & Crossan, 2019) While thethermodynamic Action is an important consideration for the transport ofentropy through a vapor medium, as we shall hereinafter reveal, thedefinition of Action and the Principle of Least Action from which itderives is best defined by Rose et. al., other than the requirement tonote that Action provides a unifying principle for Clausius' thermal andBoltzmann's statistical interpretations of entropy. (Rose, Crosssan, &Kennedy, 2008)

We consider Wu and Liu: “Current mainstream studies of the Earth'sclimate are primarily based on the principles of energy, momentum, andmass balances to develop climate models such as general circulationmodels (GCMs) to simulate various phenomena under investigation withinthe system.” (Wu & Liu, 2010)

“To improve climate models and to reduce the large range of climateuncertainties, it appears necessary to seek additional constraint(s) ofthe Earth's climate system. Investigations along this line are generallyrelated to the second law of thermodynamics wherein entropy and entropyproduction are fundamental components, and thermodynamic extremalentropy production principles are often used to explain some collectivebehaviours of the complex Earth's system without knowing the details ofthe dynamics within the system. Such thermodynamic investigations haveprovided crucial insight into various processes of climatic importancein the past several decades.

“However, the entropic aspects of climate theory have not yet beendeveloped as well as those based on energy, momentum, and mass balances.

“Furthermore, the Earth system as a whole is virtually driven andmaintained by the radiation exchange between the Earth system and space.Solar (i.e., shortwave) radiation is the origin of almost all theprocesses on the Earth's surface and above, including oceanic andatmospheric circulations, weather, climate, and lifecycles. The Earthsystem absorbs incoming solar radiation, converts it into other energyforms through various irreversible processes, and reradiates terrestrial(i.e., longwave or infrared) radiation back to space. Under a steadystate, the amount of energy emitted by the Earth system in the form oflongwave radiation is balanced by the absorbed shortwave radiationenergy. However, the emitted longwave radiation has much higher entropythan the absorbed shortwave counterpart because the temperature of theformer is much lower than that of the latter. The resulting negative netentropy flux from the radiation exchange between the Earth system andspace quantifies the rate of the Earth system's internal entropyproduction. As a measure of the overall strength of all the processeswithin the Earth system, the Earth system's internal entropy productionis an important macroscopic constraint for the Earth system in additionto the principles on which the modern GCMs are built.” (Wu & Liu, 2010)Whenever one sees an negative entropy balance, it requires an increasein entropy somewhere else in the system that is not measured or beingconsidered within the balance statement. The entropy in the GlobalClimate Model inferred by Wu & Liu is in contrast to the role of entropyin photosynthesis and the dissemination of radiation from the leafitself.

In this sense or, at least, not as a completely tenuous parallel, ANSLbased on Down-Conversion operates in a manner similar to that of theEarth system and the thermodynamic understandings of the latter may beoperable and of utility in consideration of the former.

One can learn about the interaction of radiation with gases by reviewingthe explosion of research reports on global warming. For example, weconsider, “In the atmospheric sciences, it has now been realized that aCO2 molecule, activated by . . . absorption (of infrared radiation) tovibrate more vigorously, will transfer most of this energy to other airmolecules in the next collisions, thus increasing their action andentropy while dissipating the activated internal state and increasingtheir Gibbs energy. Furthermore, the more dilute the gas (e.g., N₂O andCH₄), the greater its translational entropy, although its vibrationaland rotational entropies will be purely a function of temperature. Thus,on absorbing a specific quantum of (infrared) radiation (excitingmolecular vibration), such a dilute gas will have a largerdisequilibrium between its vibrational action and its translationalaction. In a subsequent collision, the greater inertia and amplitude ofthe vibrating atom should cause a more efficient transfer of momentum tosurrounding air molecules, irrespective of whether they are greenhousegases or not.” On the other hand, standard molar entropy is an extensiveproperty of matter (in this case gases), which means that the greaterthe quantity the greater the sum of the molar entropies. [Mass andvolume are examples of extensive properties. An intensive property is aproperty of matter that depends only on the type of matter in a sampleand not on the amount. Among the intensive properties are thosementioned hereinafter in this instant invention: chemical potential,specific heat, thermal conductivity. Other intensive properties arecolour, temperature and viscosity.]

That the source of the primary radiation in ANSL is within the device(not external as is the Sun, as in the case for solar-powered radiationdevices) is not a limiting constraint on such lighting systems as theprimary radiation may be positioned external to the device, if sodesired. The phosphor in the Down-Conversion system absorbs incomingshortwave radiation from a p-n junction and converts it into otherenergy forms through various irreversible processes and then reradiateslongwave radiation back into space. Under a steady state, the amount ofenergy emitted by the ANSL system in the form of longwave radiation(which includes heat distributed by convection or conduction in theultimate form of thermal radiation) is balanced by the absorbedshortwave radiation energy. However, the emitted longwave radiation hasmuch higher entropy than the absorbed shortwave counterpart because theradiation temperature of the former is lower (although not much) thanthat of the latter. Please note in this case that the temperature we arereferencing is not ambient but radiation temperature, which shall bemore fully described hereinafter. Whereas energy, momentum and massbalance have a cohesive understanding of, and measurement thereof,entropy has a more amorphous fundamental property and there are manydefinitions thereof. This makes a cohesive understanding of entropy fluximpact on the Earth system in climate studies and Down-Conversion inGeneral Illumination more difficult to discern.

Again, consulting Wu and Liu: (Wu & Liu, 2010)

“For example, a wide variety of expressions have been used incalculation of the Earth's radiation entropy flux. Some studies simplyuse radiation energy flux divided by the absolute temperature as themeasure of radiation entropy flux. This approach assumes a directanalogy of radiation entropy to Clausius's definition of thermodynamicentropy for a non-radiation material system. Others estimate theradiation entropy flux by making a direct analogy to the expression ofblack-body radiation entropy flux. Still others employ and approximatethe Planck “mechanical” expression of spectral radiation entropy flux.The values of the Earth's radiation entropy fluxes at the top of theatmosphere calculated from these different expressions can differsubstantially. The inconsistency of the approaches for estimating theEarth's radiation entropy flux prohibits a sound understanding of thethermodynamics of the Earth system.” (Wu & Liu, 2010)

Importantly, the entropy inconsistencies in the climate models are mostoften impacted by how each model treats the scattered radiation. Theimplementation of Down-Conversion in ANSL requires scattering of theprimary radiation to be combined thereinafter at the site where work isto be performed: the human optical organ.

The interest therefore on thermal radiation is to use it, regardless ofhow it is initially presented and or concurrently generated, to increasethe amount of radiation that is emitted within the desired frequencyrange, by simultaneously populating—by incident luminescent (primary)radiation and incident thermal radiation or incident heat—the requiredstates that leads to the two-state situation. If one defines the QuantumYield of secondary radiation (the achromatic radiation) as that which isemitted, in photon numbers, versus that primary radiation (from thesource of primary radiation only) which is absorbed, in photon numbers,then the Quantum Yield of Down-Converted secondary radiation of thedesired frequencies, can exceed a conventional theoretical limit of 100%as a consequence of the contribution of the incident thermal radiationor incident heat. More important that the Quantum Yield exceeding thetheoretical limit of 100% is that the operating system approach athermodynamically limit of 100%.

For those devices where the Quantum Yield is less than 100% efficient,the simultaneous population of the required states by 1) primaryradiation and 2) thermal radiation or heat may increase the efficiencywhile still adhering to the conservation laws of thermodynamics (firstand second laws). The essential challenge of radiation induced processesis how much work can be obtained after the absorption of light.

This is fundamentally different than an analysis of Quantum Yields whichare frequently measured under idealized conditions and are presumed tobe independent of radiation intensity. In the alternative, athermodynamic treatment and a consideration of Gibbs free energyincorporates radiation intensity, even if indirectly it is incorporatedinto the temperature of the incident radiation. There is no fundamentalreason why photo-conversion processes should not be covered by thethermodynamic laws of nature and these laws reflect in the macroscopicthe overall efficiencies and its exchange with its environment, even ifeach microscopic examination, including consideration of rates andquantum mechanical rules for allowed and forbidden processes, appear tobe well defined.

As implied hereinbefore, the measured efficiency of General Illuminationdevices normally concentrates on energy or energy-related specificationssuch as energy flux, energy density or power, as well as radiometric orchromatic analogues thereof, which shall be more fully definedhereinafter. Contrastingly, the importance of entropy of radiation inphotolytic processes, (such as that which powers Down-Conversion), orentropy transfer, is not often considered and has not previouslyinspired prior inventions, before the instant invention, for the benefitof ANSL for General Illumination.

Entropy radiation (or the entropy of radiation) is the amount of entropywhich passes in unit time through a unit area in a definite directionand every ray (pencil) passing through a medium, in a point, has its ownmonochromatic temperature. Therefore, except for a system in equilibriumbetween matter and the radiation field, at any point in a medium thereis an infinite number of temperatures, independent of the temperature ofthe medium itself. On the other hand, for the equilibrium case, there isonly one temperature which is common to the medium and all the rays ofdifferent frequencies that cross it in every direction. (Agudelo &Cortes, 2010) The notion of radiation temperature, except atequilibrium, is one fraught with discord, as the distinguished Mr.Mauzerall highlighted in the context of photosynthesis. (Mauzerall D.,2013) However, there is no doubt that radiation, an electromagneticwave, carries both energy and entropy, regardless of whether atemperature can be attendant therewith. {The association of radiationtemperature from which radiation entropy stems, as we shall explainhereinafter, emanates from the well-known thermodynamic differentialchange in entropy to change in energy, the so-called thermodynamictemperature.}

Notwithstanding the above, one can see the appeal of considering entropyof radiation when considering photo-conversion on Earth, initiated bythermal radiation from the sun and the temperature at which the processdetails takes place on the Earth (or the interface of Earth's radiationentropy fluxes at the top of the atmosphere) as two temperatures can beidentified: that of the blackbody-simulating Sun and that in which thephoto-process takes place. One recalls that the maximum work which canbe obtained from an engine (an energy converter) operating between twotemperatures and the maximum efficiency, of the idealized Carnot Cycleassociated therewith, is:

$\begin{matrix}{W = {{\oint\left( {{dQ} - {dU}} \right)} = {{\oint\left( {{TdS} - {dU}} \right)} = {{{\oint{TdS}} - {\oint{dU}}} = {{{\oint{TdS}} - 0} = {\oint{TdS}}}}}}} & (1) \\{\mspace{76mu}{W = {{\oint{TdS}} = \left\{ {\left( {T_{h} - T_{c}} \right)\left( {S_{B} - S_{A}} \right)} \right\}}}} & (2) \\{\mspace{76mu}{\eta = {\frac{W}{Q_{H}} = {\frac{\left( {Q_{H} - Q_{C}} \right)}{Q_{H}} = {1 - {T_{c}\text{/}T_{h}}}}}}} & (3)\end{matrix}$

with η being efficiency, W=work performed, T_(h) being the hottemperature, T_(c) being the cold temperature, and S_(A) and S_(B) beingthe entropy at point A and point B in a Carnot Cycle, respectively.

We reference the summarizing words and interesting perspective ofLavergne, in this regard, who reflects more on photosynthesis as aCarnot Cycle. (Lavergne, 2006) {That a similarly distinguished scholarParson has argued, Mauzerall notes, that photosynthesis cannot bemodelled as a Carnot Cycle has no bearing on the instant invention: ourintroduction of the Carnot Cycle is simply to note that there are manyways to evaluate, define or model entropy and not all yield the sameresults, not surprising considering the diversity of applications thatthe Second Law of Thermodynamics is used to evaluate. Further, thedesire to use a Carnot cycle begs the question as to what temperaturesare proper to use for a photo-conversion process.} (Mauzerall D., 2013)

Lavergne, using a photoelectrochemical cell analogy with solar radiation(thermal radiation) being the primary radiation-initiating the process,noted that while the temperature of the blackbody Sun can be reasonablyinferred, the effective temperature of the radiation which initiates thephoto-process (upon arrival at Earth), while “Fictitious”, is T_(R) andthe temperature at which the process operates is (T). In the case ofchlorophyll a, LaVergne notes that the incident radiation based onabsorption spectrum would have a required incident energy of 1.8 eV,whereas with an incident flux of 1 photon absorbed per chlorophyll a persecond, the free energy available to perform work is only 1.36 eV, a 25%loss. As LaVergne comments, “This loss does not concern the internalenergy, which is 1.80 eV in each excited pigment. It concerns the freeenergy, due to the negative contribution of the entropic term”.Increasing the intensity ten-fold would only increase the available freeenergy by 60 mV. In this model, LaVergne notes that the radiationtemperature is 1180 K (calculated using the statistical mechanicsdefinition of entropy, from the ratio of excited states to groundstates) which is that of a blackbody that would radiate the same densityof 680 nm photons as found in sunlight on the Earth, clearly afictitious number. (Mauzerall D., 2013) Further, the hot bathtemperature, T_(h), should not be meant the temperature of the Suneither. (Lavergne, 2006) Using an attenuation factor, and thetemperature of the Sun, LaVergne obtains a free energy for the samesystem now at 1.44 eV. The important point is that the free energyavailable to perform work in a photochemical process is not the same asthe energy of the excited state from which the photoreaction proceeds.

One can see that in the solar-initiated thermodynamic treatment, thework available from absorption of radiation, is sensitive to thetemperature of radiation used and that is strictly only calculable for ablackbody. This may be a serious restriction, or it may not be, in that,as we shall hereinafter explain, unless a cavity radiator is perfectlyreflecting, that cavity body tends to emulate a blackbody cavityradiator. (There is a graybody concept that is used to define less thanperfect blackbody radiators, but we shall not use it in the descriptionof the instant invention.) Another restriction to this line of reasoningmay emanate from the thought that blackbody radiation temperature is ofa consequence of an equilibrium (between thermal radiation and matterthat is not perfectly reflecting) whereas photo-conversion is clearly anirreversible process, even if steady state from a kinetic perspective.On the other hand, thermal radiation modelling seems to be compatiblewith non-equilibrium thermodynamics. Finally, we are also reminded, aswe shall further comment hereinafter, that some of Dr. Einstein'sdistinguished understandings of electromagnetic waves and photonparticles—the thermodynamics of radiation—were conceived modellingperfectly reflecting cavities but using equations derived from perfectlyabsorbing cavities.

One can perhaps approximate radiation temperature if the primaryincident radiation is monochromatic but this is normally not the case inANSL using Down-Conversion which is dependent on a rather broad primaryemission from a blue LED. What we ultimately will have to consider isthat ANSL using Down-Conversion needs to deliver General Illumination atvery high intensity for it to be useful as intended. Thus, the radiationtemperature of the incident primary radiation will be considerablyhigher than ambient (the hot temperature).

The statistical mechanic definition of entropy is what aligns thechemistry molecular details, associated with the second law ofthermodynamics, with that of the Carnot Cycle for mechanical heatengines. In other words, it is the link between macroscopic behaviourand microscopic (atomic and or molecular) processes. We shallhereinafter comment on the link between ray optics (a macroscopicprocess) and wave optics (a microscopic process).

The Kirchhoff statistical mechanics evaluation of molecular entropy iswhat associates the second law of thermodynamics for the interaction oflight with matter. The essential element is that the population ofexcited state species to ground state species (any molecule with twostates and separated in energy by hv) is dependent solely on radiationintensity and is the same for any species, regardless of moleculardetails, at a given radiation intensity.

Further and most interesting, the population associated with theincident radiation and a given intensity (a radiation temperature) isexactly the same as the Boltzmann distribution associated with a thermaltemperature. The fundamental relationship that makes this happen is,where σ is the absorption coefficient of the absorbing species with twostates k_(f) is the rate of fluorescence from the excited state of thespecies that had absorbed the primary radiation while in its groundstate: (Lavergne, 2006)

$\begin{matrix}{\frac{\sigma}{k_{f}} = {8{\pi\left( \frac{v_{0}^{2}}{c} \right)}}} & (4)\end{matrix}$

That is to say, the better the emitter (faster the fluorescent(luminescent or radiative) rate constant), the better the absorber (viacross-section) as their ratio is a constant at a given excitation andemission frequency. The requirement for this to be completely accurateis that there is no non-radiative decay that competes with thefluorescence (the emission in the form of any radiative process). Thisis an important point that we will exploit in the instant invention. Ifone looks at excitation and emission as two separate and distinct stepswithout any memory of the first when considering the second, the aboveequation would not be considered, by many, to be applicable.

We do note however that Mauzerall's thermodynamic consideration ofexcited state processes includes the following: “A simple view of thesituation is to say that once the photon is absorbed and the excitedstate formed, it has no memory whatsoever of the source of the photon:this is an irreversible process in complex molecules . . . . This alsofollows from the simple view of loss of memory on absorption.”(Mauzerall D., 2013) We call this the Mauzerall Principle: photolysis isan irreversible, not a thermodynamic, process. The instant invention isnot dependent on whether the view of Mauzerall is correct or whetherDetailed Balance is strictly adhered to in the claimed implementation,but the entropic content of paired particles may forever link theseparticles regardless of the distance they travel from each other. Onemay increase the efficiency of ANSL if the Detailed Balance is adheredto or whether the emission therefrom exceeds a balance of theabsorption.

Now let us explain why the above equation of LaVergne

$\begin{matrix}{\frac{\sigma}{k_{f}} = {8{\pi\left( \frac{v_{0}^{2}}{c} \right)}}} & (5)\end{matrix}$

is applicable to the source of secondary radiation regardless of thesource of primary radiation or even if the excited state population iscaused by heat. One radiation source can be thermal radiation producedby a blackbody: the other produced by an electroluminescent device (wecan easily choose a photonic lamp). The rate of fluorescence can bemeasured, and this is the same regardless of how the population ofexcited states is obtained. The rate of fluorescence isentropic-independent. The absorption cross-section, whether it be ofradiation from a lamp, radiation from luminescence or radiation of thethermal type, all leads to quantum jumps: either from one electronicstate to another or through stepwise quantum jumps, each of a smallermagnitude correlating with vibrational energies. The absorptioncross-section is a function of the permittivity of the material (whichis the same regardless of how the incident frequency is generated) andthe electric field of the incident radiation (also independent of howthe frequency of radiation is generated).

As Lavergne (Lavergne, 2006) notes, the Boltzmann factor that definesthe loss of free energy available to perform work with absorption oflight of a suitable radiation temperature (or with complementaryexcitation from phonons of an appropriate temperature) is

$\begin{matrix}{\frac{\left\lbrack P^{*} \right\rbrack}{\lbrack P\rbrack} = e^{- {(\frac{hv}{k_{B}T_{R}})}}} & (6)\end{matrix}$

and that

${RT}\;\ln\frac{\left\lbrack P^{*} \right\rbrack}{\lbrack P\rbrack}$

is the free energy loss or entropy gain associated with the excitation.At low radiation intensity (radiation temperature) the population of[P*] is small and the free energy loss is greatest. When the radiationintensity is high (high radiation temperature), the entropy of thetransition is near zero and the free energy available to perform work isthe greatest. This is completely a thermodynamic argumentation. However,when [P*] becomes very high, both thermodynamics and kinetics determinethe fate of the underlying assumption that there is no non-radiativedecay that competes with k_(f).

As is well-known for thermal reactions, it is the free energy of theprocess (either Gibbs free energy or Helmholtz free energy) that defineswhether an irreversible process can proceed spontaneously. One of thetell-tale signs of this is whether the reaction is endothermic orexothermic; whether heat must be provided to allow for the thermalconversion to take place; the heat being a representation of entropy.

There is no rationalization for excluding an evaluation of free energyto ascertain whether free energy and entropy impacts processesstimulated by the incidence of radiation, photo-conversion. We return toLaverne for further guidance: (Lavergne, 2006)

“The essential point is that work/free energy is not just the sum ofinternal molecular energy, but requires collective, ‘macroscopic’interactions—pressure is perhaps the most familiar example.” (Lavergne,2006)

If we consider a photo-conversion initiated from solar thermalradiation, which in itself has the maximum entropy of any possibleincident radiation, then the photo-conversion may appear to violate thesecond law of thermodynamics if the subsequent process, for example,fluorescence, does not have greater entropy than that which is incidentin the first place (unless entropy is generated somewhere else in thesystem). Since it is not possible for fluorescence to have as high aradiation entropy as any incident solar radiation, as for any givenspectrum thermal radiation has the highest possible spectral entropy,then therefore there must be some entropy generated elsewhere (in theunlikely case that the fluorescence spectrum is identical to that ofthermal radiation, thein the process would be isentropic.) A solarpowered laser by definition then, if one is possible, would have togenerate entropy somewhere else in the system and pass along same to thesurroundings.

Notwithstanding the above, as far back as 1907, the entropy of aluminescent emission line of an atom was calculated (the Balmer seriesline for atomic hydrogen, normally symbolized as H₁—generated frommolecular hydrogen H₂ in a discharge tube—at 486.136 nanometre), by thedistinguished scientist Mr. Wien, after earlier discussions of entropyof radiation by others.

All radiation transports entropy in the same manner that it carriesenergy with the exception that radiation can be designed to approachzero entropy, whereas radiation, if it exists, must have some energy,even as its intensity approaches zero. Radiation transports information:if there is no information to be discerned (measured), there is noradiation.

Having introduced the concept of radiation entropy, we note thatgenerally, with suitable construction of an ANSL, the entropy of theluminescence (the secondary radiation), in Down-Conversion, is greaterthan the entropy of the incident primary radiation (i.e.; theelectro-luminescence) as spontaneously emitted radiation is spread overall directions and thus the number of radiation modes increase.Scattered (primary) radiation also experiences an increase in entropycompared with the incident (primary) radiation but not as much anincrease as that realized with luminescence. Consequently, by its verynature, Down-Conversion of radiation increases entropy of radiation.There is a difference in the thermodynamic evaluation of radiation andmatter. The second law requires that the capacity of an isolated systemfor work must decline over time. Notwithstanding, both radiation andmatter are embodied with entropy.

To be absolutely clear, the entropy of the excited state of matter, thebody already in an electronic state ready to, and that will, luminesce,in fact decreases with spontaneous emission, whereas the entropy of thethen produced radiation field increases. With depopulation of the numberof excited states and increasing the number of ground states, thediversity of states of the luminophore decreases. Thus, the reason theentropy of the excited-state matter, once formed, decreases with theemission of a radiation field. In a stationary state (definedhereinafter), the absolute value of the entropy gain of the radiationfield is seven to eight orders of magnitude greater than the absolutevalue of the entropy decrease of the excited-state matter. Therefore, itis difficult to optically cool matter even though the radiation fieldcarries away significant entropy. In the application of optical cooling,the incident radiation is normally of the luminescence type, indeed thatwhich is referenced to be stimulated emission (i.e.; that of a laser)and the secondary emission is normally of the anti-Stokes type whichwill be hereinafter mentioned. (Ye, et al., 2017) In Down-Conversion forGeneral Illumination using ANSL, the incident radiation is normallyspontaneous emission and the secondary emission is normally of theStokes type, which shall hereinafter also be mentioned.

The aforementioned additional entropy gain to the subsequently createdradiation field due to luminescence is, in part, related to thedirection of the emitted radiation which has famously been attributed tobe governed solely by chance (whereas the direction of scatteredradiation is, according to classical mechanics, a function of thedirection of incident radiation).

The scattering used in Down-Conversion is of diffuse reflection, notspecular reflection, to maximize illuminance at the expense of luminanceat any one point of observation. By way of definition, diffusereflection is the reflection of light from a surface such that a rayincident on the surface is scattered at many angles rather than at justone angle as in the case of specular reflection. That is to say,Lambertian reflectance is desirable for General Illumination so thatLambert's cosine law (also called Lambert's emission law or cosineemission law) is operable and the luminance observed is as independentas possible of the viewing angle from the normal to the emittingsurface, if not for the fact that the LED die emission itself is highlydirectional. [In the end, ANSL based on Down-Conversion initiated by anelectrical current passed through a LED, often use tertiary scatteringcontrivances to obscure the directional emission from the LED die,itself.]

Spontaneous thermodynamic processes are those whereby the total entropyincreases. Nevertheless, spontaneous chemical reactions, whereby theentropy for the reaction itself decreases, are well known, of course,but the decrease must be offset somewhere else with a greater entropygain. As one interesting and relevant example, the entropy decrease inthe formation of glucose in radiation-initiated photosynthesis is offsetby a greater increase in thermal radiation entropy for radiation emittedby the leaf (after absorption of sunlight) and not used for glucoseproduction. (Yourgrau & Merwe, 1968) [That the entropy of the reactiondecreases is interesting at first blush in that one is converting a gasto a solid (carbon dioxide to glucose). In photosynthesis, the freeenergy generated in the electrons of the photosystems is conserved bythe subsequent conversion of relatively high entropy substances such ascarbon dioxide and water to lower entropy substances such as sugars andoils. (Rose, Crosssan, & Kennedy, 2008) The standard molar entropy ofsolids is generally lower than that of gases or vapor; in this specificcase, as well, six (6) gas molecules are converted to one solidmolecule; both the phase change and the molar change lead to a reductionof entropy in the reaction.] This is a reminder that in aradiation-initiated spontaneous processes, the entropy of exitingradiation must be considered. In the case of photosynthesis, the processuses thermal radiation, from the sun, as opposed to luminescence frommatter. Recall the comments of Wu & Liu as it relates to entropy balancein radiation from the Sun to Earth and back.

Notwithstanding the benefit of a spontaneous (also called irreversible)process in thermodynamic machines, a process that increases entropy isone in which inefficiencies are fundamentally introduced. When viewedfrom a universal perspective, there is always an increase in entropyaccompanying a defined process not at equilibrium and hence there arealways inefficiencies introduced by performing that process, mostapparent when the process is performed repeatedly. This is the essenceof mechanical thermodynamics. The more times a process is repeated, themore apparent are the inefficiencies even if each microscopic process,that defines in total an engineering solution, provides work near unitefficiency. This perspective cannot be too often repeated and emphasizesthe inefficiencies created in the macroscopic world of engineeringdevices.

For a system at equilibrium, of course, despite the fluctuations fromequilibrium that may occur, there is no change in entropy as the changeis dependent solely on the entropy of the initial and final states, theyare the same in an equilibrium. However, for a spontaneous irreversibleprocess that is clearly not an equilibrium system, an increase in totalentropy is inevitable. As an example, a non-equilibrium system whichtransfers heat or energy with its outer environment, the heat bath, thechange in entropy (D_(E) _(nt) ) with time t,

D _(E) _(nt) /Δt  (7):

is a sum of the local increase in entropy within the system and theincrease in entropy across the system boundary!

A non-equilibrium system at steady state can keep a low entropy bydiscarding high entropy fluxes across the system boundary, as noted evenif entropy also flows into the non-equilibrium system, provided theexiting entropy flux is greater than the incident entropy flux. (Kleidon& Lorenz, 2004) In the case of photo-processes, the incident radiationcan be from any type, including luminescence and thermal radiation,although the entropy content of each will no doubt be different.

The aforementioned Principle of Detailed Balance (Miller, Zhu, & Fan,2017), when considering matter's interaction with a radiation field, isa current focus in the field of solar energy conversion, (Markvart, Fromsteam engine to solar cells: can thermodynamics guide the development offuture generations of photovoltaics?, 2016); (Markvart, CountingSunrays: From Optics to the Thermodynamics of Light., 2010) from whichthe instant invention derives motivation but is distinct from:

“For example, a solar absorber absorbs light from the sun. The DetailedBalance then dictates that the solar absorber must therefore radiateback to the sun. Such a radiation back to the sun is an intrinsic lossmechanism.” (Zhu & Fan, 2014)

The principle of Detailed Balance is a fundamental property of theinteraction between matter and a radiation field: there is bothabsorption and emission of radiation and they are related: as previouslyshown, a good absorber is a good emitter. (Markvart, From steam engineto solar cells: can thermodynamics guide the development of futuregenerations of photovoltaics?, 2016) Detailed Balance does not requireabsorption of thermal radiation; it is applicable to any source ofprimary radiation.

There are three historic representations of this Detailed Balance:expressions of Mr. Kirchoff, Dr. Planck and Dr. Einstein. These havebeen summarized with great clarity, but it is imperative to note, forthe purposes of complete precision, that the rate referenced is per unitarea and not the more usual per unit time:

“paraphrased in terms of (the number of) photons rather than their(individual) energies, Kirchhoff surmised that the ratio of the rate atwhich photons are emitted by a unit area to the absorbing power “a” isequal to a universal function of only the temperature of the substanceand the wavelength or frequency of emission“, in other words, in termsof photon flux density. Dr. Planck, many decades later, “elaborated onKirchhoff's ideas by noting that radiation can only be absorbed orgenerated in a volume element of a body rather than on the surface”.(Markvart, From steam engine to solar cells: can thermodynamics guidethe development of future generations of photovoltaics?, 2016)

More than a decade later, Dr. Einstein “extended the photon balance ofDr. Planck” and showed that Planck's Law required stimulated emission inaddition to spontaneous emission in balance with absorption. Thistreatment of Dr. Einstein was initially attributed only to atoms but wassubsequently extended to molecules by the distinguished Mr. Kennard andthe eminent Mr. Stepanov, as we shall hereinafter comment.

The primary interest of the instant invention is the Detailed Balance asit applies to entropy. This approach has been the subject of manydiscussions for over one hundred years but to which no consensus exists.Employment of entropic Detailed Balance, to impact energy efficienciesin terms of photon numbers, is another main purpose of this instantinvention and heretofore not realized as being applicable to GeneralIllumination using ANSL although it has been debated in the context ofincident radiation from the Sun, a source of thermal radiation.

The entropy generated with the production of the secondary radiation isthe fundamental source of entropy gain in Down-Conversion as practicedin General Illumination using ANSL and, as earlier noted, this is offsetwith a decrease of the entropy of the excited-state of the phosphoritself, a decrease that would be greater not for the possible entropygained from the almost ubiquitous Stokes shift (discussed in more detaillater), if one is indeed present. (Stokes, 1852) This is true of allluminescence with a Stokes shift. [Indeed, the presence of Stokes shiftwas obliquely commented on by Dr. A. Einstein, in 1905, as anindication, for reasons that may be vague, of radiation displayingparticle and or wave characteristics depending on the measurementtechnique chosen to categorize or otherwise, quantitate radiation. (KuhnT. S., 1978)

For the instant invention, our interest is on the entropy of radiationregardless of the fact that radiation displays both particle and wavecharacteristics (depending on the measurement by experiment). Theimportant point for the purpose of the instant invention is that theStokes shift at the source of secondary radiation, often has beeninvoked as a source of heat and consequently as a cause of inefficiency,in Down-Conversion, is in fact where the entropy of the matter, theluminophore, is decreased as a consequence of the transfer of entropy tothe secondary radiation field to which it, the luminophore, emits.Parenthetically, if it is contrarily true that with ANSL usingDown-Conversion a Stokes shift is a source of heat, then remotelypositioning the luminophore, as is often promoted, will clearly noteliminate the heat from being generated at the source of secondaryillumination.

To depict more fully, we must adopt the quantum mechanical view ofmatter in its excited state, descriptions thereof that invoke thoseprocesses such as the Born-Oppenheimer Approximation, Franck-Condontransitions and Kasha's Rule. In the above description of entropytransport because of the Stokes shift, we are not focusing on possibleradiation-less decay from the lowest vibrational level of the electronicexcited state to the ground electronic state. This radiation-less decayoccurs at a slower rate: considerably slower than the vibrationalrelaxation that defines the Stokes shift itself. Radiation-less decayfrom the lowest vibrational level of an excited electronic state is, ifit occurs, may be a significant source of inefficiency and it ischaracterized by a Quantum Yield of secondary luminescence being lessthan 100%. Of course, it is and has been clear that implementingDown-Conversion for General Illumination is best when the Quantum Yieldof secondary luminescence is at least 100%, if not greater. Indeed,there is no fundamental law that precludes the Quantum Yield ofluminescence from being greater than 100% (e.g.; singlet fission); onlythat the conservation laws be followed.

On the other hand, if there is a fast rate of radiationless decay,k_(nr), that is competitive with secondary luminescence (fluorescence;k_(f)) at the source of the secondary emission upon which primaryradiation is incident upon, then the aforementioned fundamental equationwould not apply. In the alternative:

$\begin{matrix}{\frac{\sigma}{k_{f} + k_{nr}} \neq {8{\pi\left( \frac{v_{0}^{2}}{c} \right)}}} & (8) \\{{if}\mspace{14mu} k_{nr}\mspace{14mu}\text{>>}\mspace{14mu}{k_{f}.}} & (9)\end{matrix}$

We would not be complete if we did not again note, but in a slightlydifferent context, that even for those secondary phosphors with noradiation-less decay from the lowest vibrational level of the electronicexcited states to the ground electronic state, i.e.; for thoseluminescent systems with a Quantum Yield of 100%, it is often stated inthe scientific literature that the Stokes' shift is an inherent loss ofenergy, that heat is thus generated at the site of the secondaryemitter, regardless how far away the principal radiation is generated,and that is what makes Down-Conversion less efficient than thoseachromatic LED lamps that need not employ the Stokes shift.

An often-stated quantum mechanical description of this radiationlessdecay is summarized in a recent review of thermal stable phosphors(Tian, 2014) and follows the model produced in the tome by thedistinguished Arthur C. Cope Awardee and Wm. P. Schweitzer Professor ofChemistry at Columbia University in the City of New York, Nicholas J.Turro, Modern Molecular Photochemistry of Organic Molecules. (Turro,Ramamurthy, & Scaiano, 1965) Although the principal basis for theinstant invention is thermodynamic consideration of entropy ashereinbefore briefed, we shall also demonstrate the adherence to thesame Turro-paradigm that offers a quantum mechanical explanation for theinstant invention.

The Quantum Yield of luminescence at ambient may be 100% but may not beso at higher temperatures as experienced and when in contact with ornear a light-emitting diode die operating with the electrical inputcurrent used in an ANSL for General Illumination. Even if the QuantumYield of luminescence from the source of the secondary radiation fieldis near but slightly less than 100%, under high incident radiation fieldflux, some inefficiencies remain competing with secondary emission. Incontrast to the Stokes shift, the inefficiencies referred to in thepreceding sentence occurs after the associated shift has taken place andthe excited state from which emission is to take place is (said to be)at the lowest vibrational level of the electronic excited state.

To the knowledge of this inventor, regarding Down-Conversion in ANSL,while the difference in energy associated with the Stokes shift is oftenarticulated as manifesting heat at the location of the secondaryemitter, there is no time-resolved proof that this indeed does occur—theentropy associated therewith can be dissipated in other ways. This pointis emphasized in the instant invention as there is considerable priorart on Down-Conversion that is universally based on the Stokes shiftbeing a source of heat and consequently a secondary emitter isunavoidably a source of heat, at any intensity of incident radiation,even if the luminescence from the secondary emitter occurs with completeefficiency.

Entropy is conventionally described with the letter “S” which we shallherein adopt as well. The general consideration is (where “ex states”means excited states)

S _(incident(primary)) ±S _(spectral shift) +S _(population ex states)−S _(depopulation ex states) −S _(secondary radiation) ±S_(surroundings)≥0  (10):

If the transfer of entropy to the exiting luminescing radiation field isgreater than the entropy generated by the Stokes shift, then the site ofDown-Conversion (the phosphor) would have a net reduction in entropy,considering no other factors. Alternatively, if the transfer of entropyto the exiting luminescing radiation field is less than the entropygenerated by the Stokes shift, then the site of Down-Conversion (thephosphor) would have a net increase in entropy, considering no otherfactors. Nevertheless, and notwithstanding the above, it is stillunclear whether the entropy carried away by exiting radiation may beparticularly useful for work for General Illumination when same isincident upon the visual pigments of a human observer unless there issome partial re-conversion; i.e., extracted from the radiation incidentupon the visual pigments. (Zheng, 2014), (Ala-Laurila, 2003)

We further note that if the entropy received by the visual pigments of ahuman observer reflect all of the available information of the system(s)that created the radiation field, then embodying luminescence with thethermal radiation that describes the environment in which the radiationwas created, is of substantive benefit to the receiver of saidinformation, even if not in the form of illumination or luminance. Humanvision seems to be optimized for the peak maximum amount of entropycarried by the incident radiation as opposed to peak maximum amount ofenergy.

This is the fundamental conundrum of Down-Conversion and its mostindispensable understanding: while there is no proof that the energydifference of the Stokes' shift in Down-Conversion is directlydissipated by heat through convection or molecular conduction, asopposed to thermal radiation or through luminescence, the net effect onefficiency might be the same. The entropy dissipated through radiationand consequently “radiative entropy is a measure of unavailable workthat cannot be extracted” from radiation. “The goal is to improve theperformance of energy conversion systems, namely, increasing the amountof useful work and reducing the unavailable work.” (Zheng, 2014) On theother hand the Stokes' shift is relatively small, especially comparedwith heat that may be the form in which entropy is dissipated as aconsequence of radiation-less decay through internal conversion. Thus,the consequence of entropy generation via the Stokes' shift is unlikelyto be a significant loss of efficiency, if any loss is indeed the case,and, as detailed herein, an increase can be obtained through an increasein photon numbers.

It is this entropy transfer which generally restricts the spectralquantum selection rules for radiative electronic transitions, so wellexplained by quantum mechanics applied to chemical matter, and as aparallel to what was long ago called the Einstein's law of photochemicalequivalence. (Allmand, 1926) (Einstein, On the Quantum Theory ofRadiation, 1917) (Murkerji & Dhar, 1930) (Einstein, ThermodynamischeBegründung des photochemischen Äquivalentgesetzes., 1912) There are manyexceptions to this equivalence law so that it is not a universal law ofnature, despite its often-debated adherence in the debate of theconception of Down-Conversion.

We initially defined luminescent states to be those which subsequentlyand spontaneously produce radiation (an electromagnetic field) of thesame or different magnitude in energy (frequency, or wavelength, orwavenumbers). Luminescent states can emit at an energy higher than thatof the primary incident radiation: this is called an anti-Stokes shift.(Pandozzi, et al., 2005) Luminescence with an anti-Stokes shift requiresthe increase of radiation entropy to more than offset the decrease inentropy associated with the luminescing matter (due to depopulation ofexcited states) as well as the decrease in entropy associated with theheat bath at temperature T that otherwise provides the additional energyof the anti-Stokes shift. As we shall hereinafter note, and to clarifythe interrelationship referenced in the preceding sentence, entropy whenmultiplied by the temperature is in units of energy.

While hereinbefore we have discussed luminescence in a general manner,and the effect of entropy on same—not ignoring, of course, theimportance of energy nor momentum—Down-Conversion to generate achromaticlight has an additional consideration to be dealt with. Focusing onentropy of radiation that reduces efficiency, achromatic Down-Conversion(the combination of scattered primary radiation with secondaryradiation) experiences an otherwise deleterious entropy of mixing, evenif the two sources are monochromatic (provided they are notinterfering). (Laue, 1906) (Zheng, 2014) That being the case, theentropic contribution to the process is spontaneous (regardless of theenthalpy of the radiation from the two sources) and the outgoingradiation entropy is greater than the sum of the radiation entropy thatemanated from each of the two sources absent the mixing. This is oneunavoidable origin of the inefficiencies of Down-Conversion associatedwith mixing of the light necessary to perform work: as an example, themixing of thermal radiation from two blackbodies results in an increasein entropy of the combined radiation. (Khatri, Sunyaev, & Chluba, 2018)This is true whether the radiation fields are in equilibrium with theirrespective matter, or not. (Khatri, Sunyaev, & Chluba, 2018)

Prior to the disclosure filed on Mar. 26, 1996—within U.S. Pat. No.6,600,175—almost all other attempts to prepare achromatic radiation frommultiple light emitting diodes sought achromaticelectroluminescence—broad-based but identical emission from each of thediodes, themselves. (See as just one, among many, examples, thedisclosure within JH09232627-1997-09-05—Hidemi, Yuzaburo, Isao, andAkihokp—“to provide a white LED of high purity and reliability” . . .with “three active layers, different from each other in band gap energy,are provided” and “light rays emitted from the active layers are lightof three primary colours, red light, green light, and blue light” . . .and “only white light is emitted”.) No doubt the general presumptionwas, prior to '175, that the inefficiencies introduced by the entropy ofmixing three different frequencies of radiation, from three differentcolored LED devices (to make white light), would be too great to afforda lamp appropriate for General Illumination and hence the preference fora single electroluminescent device emitting all of the required coloursto appear white to a human observer. (U.S. Pat. No. 5,136,483, 1992)

However, we are mindful of a perhaps different perspective, therepresentation of Lavenda, referring to a thought experiment of Lorentz,that the splitting of pencils of radiation of the same frequencies intoreflected and transmitted rays. Therein it is said that there is a gainin entropy as a consequence of the splitting of the rays and that thereverse process, mixing the before split rays, would not be spontaneousas it would require a reduction in entropy. (Lavenda, 1991)

Notwithstanding the great progress in ANSL (LED) lamps for GeneralIllumination since '175, understanding the source of currentlyrecognized inefficiencies and ameliorating, mediating, modifying oreliminating same is of societal benefit to improve the end-user'sacceptance—of commercial achromatic light-emitting diodes—as instrumentsfor General Illumination and a concomitant reduction in energy usageassociated therewith.

The present-day commercial success of white LEDs was caused by thesubject matter of the aforementioned patent (i.e.; '175), as distinctfrom the prior art at its time of invention. The disclosure within '175emphasized as a preference (a preferred embodiment), for the source ofsecondary radiation, incorporating “fluorescent materials with extremelyshort radiative lifetimes, less than 50 nanoseconds”, effectively a rateof luminescence fast enough to ameliorate the inefficiencies associatedwith using higher intensity primary radiation sources.

Intensity dependent studies have recently been performed in systems thatare based on the implementation of Down-Conversion to generate whitelight for General Illumination. Specifically we incorporate by referenceinteresting work on Ce³⁺:YAG phosphor luminescence estimated by highintensity blue laser diodes. (U.S. Pat. No. 7,151,283, 2006) (U.S. Pat.No. 6,936,857, 2005) (Yuan Yuan, 2018) The phosphor reported thereindemonstrated a reduction in fluorescence yield as the intensity of theincident primary radiation increased. More specifically, the authorsnoted “with the increase in the exiting laser power, the emittingluminous flux decreases”. (Yuan Yuan, 2018) This may appear, at first, adirect contradiction to the thermodynamic argumentation that a highintensity, the radiation temperature is at a maximum and that entropicdiminishment of free energy is at a minimum. However, a contradiction itis not. Thermodynamically the above conditions provide the maximumefficiency of work; but it does not define what work will be performed.Clearly rates of reactions determines competitive yields and at highintensity, and maximum work and high concentration of excited states,rates competitive with k_(f) come to dominate. (Mauzerall D., 2013)

We shall hereinafter define the radiometric terms more completely, butit is beneficial to note that Radiant Flux (also called Radiant Power)in Joules per second or in Watts is a measure of Radiant energy emitted,reflected, transmitted or received, per unit time. This is sometimesalso called “radiant power”. As we hereinbefore noted, the temperatureof radiation at a frequency (energy) is in fact strongly influenced byits intensity (power or flux); the higher the intensity, the higher theradiation temperature. As we previously summarized, at low radiationintensity (radiation temperature) the population of [P*] is small andthe free energy loss (with each additional absorption) due to thestatistical mechanical entropy term is greatest. When the radiationintensity is high (high radiation temperature), the entropy of thetransition is near zero and the free energy available to perform work,as a consequence of the transition, is the greatest. This is completelya thermodynamic argumentation. However, when [P*] becomes very high,both thermodynamics and kinetics determine the fate of the underlyingassumption that there is no non-radiative decay that competes withk_(f). Indeed, for General Illumination using Down-Conversion in ANSL,the rate of non-radiative decay due to k_(nr)[P*] becomes competitivewith the rate of fluorescence, k_(f). Thus, while overall work is at itsgreatest, that work indeed leads to an increase in processes other thanfluorescence, as statistical thermodynamics does not favour one processover another.

The decrease in yield in fluorescence with increasing intensity ofincident radiation is one of the perhaps surprising reason why it wouldnot be generally expected, or universally predicted, that increasingpower of emerging blue GaN light-emitting diodes would necessarilytranslate into increasing white light illumination unless there was someunique means to accomplish same. In other words, that more initial(primary) photons might be available, and incident on a phosphor, wouldnot necessarily mean that the efficiency of down-converted lightemission by the phosphor would not necessarily decrease. Thus,intensity-dependent inefficiencies would mean that the advantage ofbrighter emission by the phosphors would not be necessarily aconsequence of the brighter incident primary radiation. Indeed, onemight argue that this is clearer that the converse, despite the opinionsof many which are apparently based on misconception of and betweenenergy and intensity (or flux).

Singlet-singlet annihilation, which occurs at high radiationintensities, is detrimental to operation of high brightnesslight-emitting devices. (Ruseckas, et al., 2009) (Barzda, et al., 2001)(Nettels, et al., 2015) (Jordens, et al., 2003) (King, 2008)Notwithstanding the above and as disclosed in '175: 1) very highradiation intensities of the primary source were required to generateenough illuminance for useful work and 2) such primary radiant sourceswere available with the introduction of “high-brightness” blue LEDs ofhigh luminance, based on the work of many.

However, it was generally believed at the time of the inventiondisclosed in '175 that such high-brightness sources could not be used togenerate white light via Down-Conversion (as evidenced by disclosure ofcontemporaneous thinking “we did not believe that it was possible toproduce white light in a similar manner”) despite the prior art that wasable to Down-Convert using less than “high-brightness” light emittingdiodes to convert violet light-emitting diodes to blue light emittingdiodes using a “fluorescer”; i.e.; the “similar manner” explicitlyreferred to. [Nakamura, S. (2014). Declaration of Dr. Shuji Nakamuraunder 37 C.F.R. § 1.132. United States Patent & Trademark Office ControlNo.: 90/013,225; signed on Nov. 20, 2014.]

Subsequent discussion on the matter, many years later, concluded thatbecause “Nakamura's LED . . . would provide more photons to bedown-converted by the phosphors and thereby provide brighter overalllight emission from the device”—parenthetically an assumption that thereferenced author must view as reasonable—the advantage of brighteremission by the phosphors would be readily a consequence of the brighterincident primary radiation. This statement is based on the simpleassumption that so-called “high brightness” incident radiation,expressed as numbers of photons as opposed to the total energy ofphotons, will lead to greater number of exiting photons from thephosphor as both scattered primary and exiting secondary radiation isrequired. This is essentially an argumentation of photon density.Important to the instant invention is the difference between thepractical technology of General Illumination (normally operated at highphoton density and high concentrations) and the photophysical science ofluminescence (normally examined at low photon density and lowconcentrations).

The latter, the photophysical science of luminescence—as described inthe often-referenced text “Fluorescence and Phosphorescence” by PeterPringsheim (Pringsheim, Fluorescence and Phosphorescence, 1949)—isessentially a primer on the quantum mechanical nature of matter asevidenced by its interaction with a radiation field, understanding ofwhich requires elimination of competing pathways of activation anddeactivation, generally eliminated by experimentation at low photondensities and low concentrations of matter. The former, GeneralIllumination practised at high photon density and high concentrations,is an engineering discipline, like the roots of thermodynamics (e.g.;steam engines), that is practiced in an environment where competingpathways of activation and deactivation are present as a consequence ofthe fundamental macroscopic manner in which mechanical devices mustoperate to provide the principal engineering benefit: practical GeneralIllumination at useable and generally high intensity (luminance orilluminance).

At such high levels of excitation, however, the concentration ofphosphor excited states increases dramatically to levels where thestatistical mechanic entropy nears zero, providing maximum free energyto perform work, whether useful (desired) or not. At such high levels,the work performed because of the incident radiation includessinglet-singlet annihilation, a highly probable bimolecular (two-body)process when entropy is near zero, unless the singlet excited stateshave relatively short radiative lifetimes. This mechanism forannihilation, which in the case of General Illumination wastes freeenergy, is reminiscent of the mechanism for three-level and four-levellasers. The engineering description is frequently referenced assaturation. A similar problem occurs from an engineering perspective atthe light-emitting diode used in General Illumination, itself. There thesaturation is called a “droop”.

Using the nomenclature that Y₁ represents the singlet excited state ofthe secondary emitter, such as Ce³⁺:YAG, then singlet-singletannihilation can be expressed as Y₁+Y₁→Y₀+Y₂ (where Y₀ and Y₂ representground electronic state and higher than the first singlet excited state,respectively) whereas stimulated emission is represented by Y₁+hv→Y₀+2hvand multi-photon absorption is represented by Y₁+hv→Y₂. Thesinglet-singlet annihilation is of the Forster Resonance Energy Transfertype and does not require the two Y₁ states to be in contact. In adensely populated phosphor, a possible mechanism of Forster ResonanceEnergy Transfer Y₁+Y₀→Y₀+Y₁ allowing for excitation hopping to thenearest secondly excited Y₁. (Borisov, Gadonas, Danielius, Piskarkas, &Razjivin, 1982) (Hofkens, et al., 2003) (Volker, et al., 2014) Tell-talesign of the onset of singlet-singlet annihilation is the emission fluxreduces with increasing incident flux, the radiative lifetime decreaseswith increasing incident flux, and the spectral distribution becomesmore blue, as if the phosphor experienced an anti-Stokes shift. Indeed,when the quintessential ANSL Down-Conversion phosphor, Ce³⁺:YAG isexposed to ever increasing incident radiation flux, from a blue LEDlaser, the exiting flux after the Stokes shift, from the phosphor,decreases. This—higher incident brightness, lower exiting flux—is anexcellent example of how at higher incident flux, thermodynamically,processes become more efficient and unwanted processes that stillperform work, although unwanted work, can compete and are observed tocompete against those wanted processes (in the case of GeneralIllumination, Down-Conversion to yield achromatic radiation).

Since the Down-Conversion for General Illumination does not incorporatean optical resonator, the relative amount of stimulated emission fromthe phosphor is minor, and the macroscopic population of phosphorexcited states interacts with other phosphor excited states(matter-matter interaction, as opposed to photon-matter interaction).Indeed, it had been, contemporaneously to '175, posited, in oppositionto the Down-Conversion approach, that to generate white light from lightemitting diodes, the combination of three separate lamps (one, red, onegreen and one blue) would be a successful approach (similar to the workof the hereinbefore referenced Hidemi, Yuzaburo, Isao, and Akihokp). Ofcourse, at elevated temperatures, the entropy of mixing three separateand identifiable radiation sources would be even greater than theentropy of mixing blue and yellow radiation sources leading to greaterinefficiencies associated with the former than the latter, all otherfactors being equal.

That excited singlet states of chromophores act as mobile quenchingcenters for other excited singlet states, has been demonstrated inaggregated and connected systems with many chromophores. Suchsinglet-singlet annihilation indeed does lead to decreased excited-statelifetimes at high excitation intensities and concomitantly to a decreaseof the fluorescence yield. (Barzda, et al., 2001)

Thus, the reason why the '175 disclosure was different than the priorart (and contributed to the commercial success of white light LEDspost-'175) was the realization that: 1) high levels of white lightilluminance was required for the practical engineering of GeneralIllumination; 2) to obtain such high levels of white light usingDown-Conversion, one needed to use primary LED sources of highbrightness; 3) to avoid the inevitable wasting of such high brightnesssources due to the entropic-reality that (as a consequence of highconcentration of singlet excited states and quenching processes such assinglet-singlet annihilation in the source of secondary radiation)radiative lifetimes of less than 50 nanoseconds were preferredembodiments. Indeed, most of the phosphors used in commercial LED lampshave radiative lifetimes near 50 nanoseconds or less (as referred by'175) and most definitively less than 100 nanoseconds.

The shorter radiative lifetimes in the preferred embodiments representthe reciprocal of pure rates of luminescence that must not be madeshorter due to quenching effects, whether by external atoms or byself-quenching. As by example to demonstrate this point, the phosphorCe³⁺:YAG is frequently used in Down-Conversion of blue radiation emittedfrom a GaN light-emitting diode whereby, in at least one preparation ofthe phosphor, by He, et. al., two radiative lifetimes were observed forthe luminescence perceived to be yellow absent the mixing with scatteredblue radiation: 26.5 nanoseconds and 91.1 nanoseconds. (He, et al.,2016) However, the shorter radiative lifetime of the luminescent ion wasattributed to self-quenching (normally associated with increasingconcentration) and hence not a pure reciprocal rate of radiationobtained at infinite dilution. The longer radiative lifetime reported inthe referenced work is admittedly longer than the 60 to 70 nanosecondradiative lifetime for conventional Ce³⁺:YAG phosphors traditionallyused in white light Down-Conversion: the longer radiative lifetime mostlikely is related to energy hopping, of the Förster Resonance EnergyTransfer type, which is known to occur with increasing concentration.The mechanism of self-quenching was not explicitly articulated. As wementioned hereinbefore, that mechanism can be singlet-singletannihilation (shorter lifetime) or Förster Resonance Energy transferenergy (longer lifetime) hopping or both (both shorter and longerlifetimes than usual). A third possible mechanism is Y₁+Y₀→Y₀+Y₀+heat,which calorimetry could be used to confirm. Consequently, it is probablethat the two different lifetimes are an outcome of two separatephenomena: radiation imprisonment which leads to longer observedlifetimes—energy hopping—and singlet-singlet annihilation which leads toshorter observed lifetimes. (The referenced work notes FT-IRspectroscopic peaks at around 790 cm-1 which equates to 12.66μ and 2.26kcal/mol and another spectroscopic peaks around 3432 cm-1 which equatesto 2.9μ and 9.8 kcal/mol.) (He, et al., 2016)

It has been recently suggested that laser diodes will be used to powerwhite light LEDs. No doubt these systems will benefit from phosphorswith radiative lifetimes even less than 25 nanoseconds and it will bepreferential to use phosphors with radiative lifetimes less than 5nanosecond (less than 5,000 picoseconds), otherwise the observation ofsecondary luminescence saturation will become critical.

For a luminophore, whether a phosphor or not, the shorter the radiativelifetime is measured with greater precision, the more indeterminant theenergy of the excited state is (as energy and time are conjugatevariables) where

$\begin{matrix}{{{{\Delta\; E\;\Delta\; t} \geq {\frac{\hslash}{2}\mspace{14mu}{or}\mspace{14mu}\Delta\; X}} = {{\Delta\; A\;\Delta\; B} \geq {\frac{1}{2}{{< {\psi{\left\lbrack {A,B} \right\rbrack }\psi} >}}}}}{and}} & (11) \\{{\Delta\; X} = \sqrt[2]{< {\psi{X^{2}}\psi} > {- \left\langle {\psi{X}\psi} \right\rangle^{2}}}} & (12)\end{matrix}$

is the standard deviation from measuring |ψ with the observable X)(Wehner & Winter, 2010); the broader the emissive spectral line(homogeneous broadening), and the greater the entropy gain in theradiation field as a consequence of the resulting spontaneous emission.(Vallance, 2017) (Weinstein, 1960)

Consequently, provided the radiative lifetime is short-enough, theentropy gains due to this homogeneous broadening into the radiationfield with secondary emission offsets to a full extent the entropyassociated with the Stokes' shift at the site of secondary emission.

In other words, for those systems that provide for General Illuminationincorporating a Stokes' shift, the means of removing theentropy—otherwise generated at the source of secondary emissions—throughradiation is enhanced by using luminophores with extremely shortradiative lifetimes. Dissipation by internal conversion and thenmolecular vibrations for nonradiative decay from the lowest vibrationallevel of the electronic excited state—internal conversion is the ratelimiting step, not the subsequent molecular vibrations—has too slow arate to compete.

A consequence of the previously mentioned preferred embodiment in '175,which is incorporated herein by reference, is that the inefficiencyassociated with entropy generation as a consequence of the Stokes'shift, if indeed this is happening without recourse as we shallhereinafter further comment, leads to an increase in dissipation of theentropy by the exiting radiation field. It is important to note that thepresumption that vibrational relaxation rates in an excited electronicstate is the same as the vibrational relaxation rates in a groundelectronic state, once internal conversion to the ground electronicstate is effected, is frequently articulated to be a consequence ofvibrational energy transfer to the plethora of nearby solvent moleculesin which the luminophore is solvated. However, Down-Conversion forGeneral Illumination as herein described is normally not constructedwith a fluid surrounding the source of secondary radiation. Thus, thefundamental consideration of dissemination of entropy from the firstelectronic excited state; the rate of internal conversion followed byvibrational relaxation (k_(v.r) ^(i.c.)) would have to compete witheither the rate of thermal radiation (k_(t.r.)) or the rate forluminescence (k_(f)) to carry away the entropy, with respect to theStokes' shift.

The applicable rate equation then is

k _(v.r.) ^(i.c.)/(k _(v.r.) ^(i.c.) +k _(t.r.) +k _(f))  (13):

where the entropy gained due to the Stokes' shift remains embodied withthe matter that is the source of the secondary radiation until somethingelse happens. The general assumption related to the Stokes' shift isthat the vibrational relaxation is irreversible and that the heat isdissipated by conduction and then convection.

This is not possible, however if the same vibrational bands arepopulated by an external and different radiation sources. Further, in asolid-state implementation, such as proposed here, in this instantinvention, conduction away of the entropy in the form of heat to theenvironment is not likely to take place, especially if the environmentis created in a manner that prevents, mitigates or otherwise reducesconduction, which is another primary purpose of the instant invention.Indeed, there is an excited state equilibrium (Demars, 1983) (Maus &Rettig, 2002) setup in that the entropy associated with the Stokes shiftis initially transferred from the chromophore, in whatever form thismight take (e.g.; Ce³⁺) to nearby phonons (e.g.; of the YAG lattice) andthen back again, unless the entropy is somehow otherwise dissipated(e.g.; from the YAG lattice). If we denote Ce_(**) ⁺ as a chromophore ina hot electronic excited state (Lin, Karlsson, & Bettinelli, 2016) andYAG_(**) as a phonon excited lattice, then the equilibrium is

$\begin{matrix}{\left\lbrack {A\overset{def}{=}{Ce}_{**}^{3 +}} \right\rbrack{\frac{\overset{k_{AB}}{\rightarrow}}{\underset{k_{BA}}{\leftarrow}}\left\lbrack {B\overset{def}{=}{YAG}_{**}} \right\rbrack}} & (14)\end{matrix}$

and the excited state equilibrium constant is

$\begin{matrix}{\frac{\lbrack B\rbrack}{\lbrack A\rbrack} = {K_{eq} = {\frac{k_{AB}}{k_{BA}}.}}} & (15)\end{matrix}$

As with all equilibriums, excited state equilibriums allow for themeasurement of thermodynamic parameters. As an example, the excitedstate equilibrium between two rotational conformers of stericallyrestricted Donor-Acceptor Biphenyl, which ultimately is depopulated fromthe excited by a considerable slower fluorescence from both conformers,yields change in enthalpy of −0.59 kcal/mol and a change in entropy of−0.7 Joules per mol K. The change in activation energy going from theexcited state of one conformer to the other is approximately 3.5kcal/mol.

The current inventor is aware that some would argue that speaking ofthermodynamic equilibrium over such short time scales is of littlecredibility, as one often sees articulated when consideringphotosynthesis, where the source of incident radiation is thermal, fromthe sun, and the entropy of the incident radiation is easy to calculate,as we shall hereinafter explain. Nevertheless, if we note that the rateof fluorescence (in reciprocal seconds) from typical phosphors used inDown-Conversion for General Illumination is near 10⁹ and that ofvibrational relaxation of the excited state due to the Stokes shift isfive orders of magnitude faster (10¹⁴), that spread is hypotheticallyequivalent to an equilibrium that lasts for at least 100,000 days (274years) if the original equilibrium had been reached one day afterreaction.

It is important to remember that formally, the rate of thermal radiationdefined hereinabove reflects thermal radiation from the groundvibrational level of the first electronic excited state. Normally oneconsiders thermal radiation as occurring from matter and is dependent onthe temperature of that matter and does not identify itself as comingfrom any one particular electronic state. The essential observation ofwhich the instant invention is dependent is that a two-level atom's (ormolecule's) luminescence emission carries along with it the thermalradiation so that the underlying luminescence carries with it theentropy of the thermal radiation which would emit if there was noluminescence and no non-radiative decay, the latter of which isinitiated by internal conversion. What this means is that there can bestimulated emission of thermal radiation and that thermal radiation istransmitted co-incident with the luminescence if the two-level statesare surrounded by an environment in which the entropy resides (e.g.; theYAG lattice with its Ce³⁺ associated chromophore).

Going back to consideration of the excited state equilibrium, Demas soeloquently notes, “(An) an important case where an excited stateequilibrium is suitable (,) arises in thermal equilibration of theexcited state manifold. For example, in condensed media, equilibriumwithin the excited-states singlet manifolds is typically complete withina few picoseconds following excitation to upper (excited) singletstates. For an emission occurring on a nanosecond time scale,equilibration of the emitting manifold can be considered essentiallyperfect.” (Demars, 1983)

In the above example, one might consider over a very long period of time(even past the typically nanosecond time period of luminescence toperhaps days or years) a steady state approximation as withDown-Conversion for General Illumination, the primary radiation isalways generated as long as the current is directed through the sourceof primary radiation and, assuming no luminescence saturation, there arealways enough ground states at the source of secondary radiation toreceive and act when primary radiation impinges. The steady stateapproximation is only exact, however, if either the Ce_(**) ³⁺concentration is much greater than the concentration of YAG_(**), orvice versa, at any one point in time. One way to ensure that the steadystate approximation is not operable and that the excited stateequilibrium is dominant is to simultaneously populate YAG_(**), andCe_(**) ³⁺ by irradiation with an infra-red light LED and a blue lightLED, respectively. With equilibrium at play, standard thermodynamicfunctions can be used to describe the states, their processes andevaluate underlying efficiencies.

In the event that there is more than one phonon B

YAG_(**) to be activated by an independent infrared LED with frequencyv, then when one might populate B′

YAG_(**), if you will, with frequency v′. It is then clear that forevery frequency that is characteristic of a phonon of excited stateYAG_(**), one might have not only the source of primary radiationactivating the Ce_(**) ³⁺ chromophore, but also the plethora ofYAG_(**), phonons. Consequently, one has by virtue of sources of visibleand infrared radiation sources, contrived a simulated blackbodyradiator. As long as not perfectly reflecting, as we shall hereinafterexplain, the system will be in equilibrium, there being no differencebetween excited state equilibrium and equilibrium, as the number ofphonons that are available and which are populated approaches infinity.At infinity of course, the system will no longer have thecharacteristics of Ce³⁺:YAG and in fact will have an emission spectrumindependent of any one material used to construct the system. Regardlessof whether the system is only at an excited state equilibrium or a totalequilibrium, we are reminded that the expressions energy and entropy atequilibrium developed by Planck (Planck & Masius, The theory of heatradiation (Translation)., 1914) and others (Bose, 1924) (Einstein, Übereinen die Erzeugung and Verwandlung des Lichtes betreffendenheuristischen Gesichtspunkt., 1905) for blackbody radiators has alsobeen shown to apply to non-equilibrium cases. (Rosen, 1954) Boroskinaet. al. teaches that “the introduction of the concept of the photonchemical potential enables establishing relations between the number ofphotons and the number of other quasiparticles photons interact with.These interactions can be modelled in analogy with chemical reactions(i.e., equilibrium is reached when the sum of the chemical potentials,including that of photons, is zero).” (Boriskina, et al., 2016)

The number of photons present are generally unrestricted except asimpacted by those particles in which the photons interact (i.e.;conservation of momentum; conservation of energy). Boroskina summarizeswith two important observations: 1) photons with a positive chemicalpotential carry higher energy per photon state than thermally emittedphotons at the same emitter temperature; 2) photons with negativechemical potential which are emitted, carry less energy than the photonsin the blackbody spectrum. (Boriskina, et al., 2016) [It is noted thatBose's derivation of Planck's formula is inherently based on the quantumand momentum and thereby avoids classical theory used by hispredecessors to derive the formula: “As opposed to these (otherderivations), the light quantum hypothesis combined with statisticalmechanics (as it was formulated to meet the needs of the quantum theory)appears sufficient for the derivation of the law independent ofclassical theory”.] (Bose, 1924)

Lasers have been previously used to generate chromatic light throughDown-Conversion (Pinnow). (Van Uiteret, Pinnow, & Williams, 1971) Lasershave near-zero entropy in their radiation field. If one viewsDown-Conversion as the transfer of radiation from one surface (the GaNLED die) to another (the phosphor), a thermodynamic treatment for theradiation energy emitted by the second surface depends only on itstemperature and emissivity but the entropy emitted by the second surfacealso depends on the entropy of the incoming radiation. (Zhang & Basu,2007) This is a fundamental difference between energy transfer andentropy transfer, both of which occur with the interactions of radiationand matter. A colloquial way of so saying, the post-transfer matter maynot be characteristic of the source of radiation energy, but thepost-transfer matter will always be representative of the source ofradiation entropy. This is the fundamental reason why the source ofprimary radiation is important in the Down-Conversion process whichadmittedly defies conventional, if not erudite, wisdom.

Retrospectively it is now realized that the lower the entropy of theincident radiation, the greater the effectiveness of the process ofDown-Conversion for General Illumination as originally disclosed in'175. Hence, polarized incident radiation, such as from light emittingdiodes that emit polarized primary radiation, (Matioli, et al., 2012)should provide increased thermodynamic efficiencies in Down-Conversionfor General Illumination, all other parameters being equal. (Zhang &Basu, 2007)

Nevertheless, even with very short radiative lifetimes, the process ofusing Down-Conversion introduces other inefficiencies that are moredifficult to circumvent. The before mentioned inefficiency associatedwith entropy of mixing increases with increasing temperature.Down-Conversion, as herein defined, realizes reduction in efficienciesas the temperature—where the mixing takes places—increases. In theorythen, Down-Conversion to deliver achromatic radiation is most efficientas the temperature at the site in which mixing takes place approaches 0K. Indeed, this is the essence of thermodynamic treatment forluminescence: only at 0 K is all of the energy of absorption (thatresults in spontaneous emission) available as free energy to performnon-heat or work that is useful. As we shall hereinafter disclose, toreduce the inefficiencies associated with the entropy of radiationmixing, one would normally expect a prudent action is to lower theoperating temperature at which the emissions from the primary radiationsource and the secondary radiation mix their spectral energy. However,the present invention does the opposite, and this is its essentialnovelty: it seeks to benefit from characteristics of thermal radiationentropy at elevated temperatures to increase the efficiency ofluminescence.

This approach is extremely counter-intuitive in that the most favouredphosphor used in Down-Conversion, Ce³⁺:YAG, is widely reported toexperience a reduction in luminescence yields with increasingtemperature, although the mechanism for same is not well explained.(Agarwal, et al., 2017) Our interest in using phosphors of the YttriumAluminium Garnet type is that in reality the intrinsic quenchingtemperature is much higher than conventionally articulated and thephosphor can be put into a protecting environment to negate many of theancillary effects of higher operating temperature. Indeed, it hasrecently been reported that the “intrinsic quenching temperature of theCe-luminescence is shown to be very high (>680 K). The lower quenchingtemperatures reported in the literature have been cautiously explainedto be thermally activated concentration quenching (for highly dopedsystems) and the temperature dependence of the oscillator strength (forlow doping concentrations).” (Bachman, Ronda, & Meijerink, 2009)

The instant invention is based on luminescence emission increasing inphoton number, to the extent that conservation laws so allow, withincreasing localized temperature, as the chemical potential of theluminescence decreases towards zero and contrary to the generalinterpretation of Einstein's law of photochemical equivalence. Thedriver for the invention is the now realized understanding that forthermal radiation there is no conservation of photon numbers and thatluminescence that looks more like thermal radiation will inheritproperties of thermal radiation as the chemical potential of theluminescent states decrease towards zero.

For the instant invention, the increasing localized temperature at thesource of secondary radiation is created either by preferentially notdissipating (by convection or by conduction) naturally produced heatthat impacts the site of, or the source of, the secondary radiation orby virtue of incorporating into the lamp, a thermal radiation source,also called a tertiary radiation source, whose spectral entropic maximumis in the non-visible, measured as if the tertiary radiation source is aperfect blackbody, and whose thermal radiation field is incident on thesource of the secondary radiation. As an example of the latter, athermal radiation source, is a mid-infrared InAs diode on InAs substratewith an InAsSbP window from Prolinx, Tokyo, Japan. The term infrared ischaracterized by the following nomenclature, at least as used in theinstant invention:

Near-infrared: from 0.7 to 1.0 μm;

Short-wave infrared: 1.0 to 3 μm;

Mid-wave infrared: 3 to 5 μm;

Long-wave infrared: 8 to 12, or 7 to 14 μm;

Very-long wave infrared (VLWIR): >14 μm.

The primary radiation source is a light-emitting diode that is bestdescribed as emitting primary radiation that appears to a human observeras being blue, but which could also emit light that is perceived to beviolet, or is ultraviolet, but which is higher in emission energy thanthe secondary radiation source. The secondary radiation source is aluminophor that absorbs the primary radiation from the primary radiationsource and emits radiation that is luminescence and thermal and which iscalled secondary radiation. The tertiary radiation source is one thatemits primary light that is not perceived by a human observer as beingvisible absent a thermal imaging device (e.g.; night-vision goggles) andwhose radiation is incident upon the secondary radiation source. Theprimary radiation and the tertiary radiation both of which are incidentupon and absorbed and or scattered by the secondary radiation sourcehowever in the case of the tertiary radiation is principally absorbed bythe source of secondary radiation. The approach of the instant inventionis different than that which may be inferred from Soares et. al. wherebyat temperatures far below ambient (12 K to room temperature), theyobserve an overall increase of the integrated intensity which was foundto be accounted for a thermal activated process described by activationenergies of 10 meV and 30 meV for the single crystal and target,respectively. (Soares, Ferro, Costa, & Monteiro, 2015)

Optimally, if an infrared (IR) light emitting diode is to be used, onewould then select such a tertiary radiation source that corresponds toan infrared absorption peak of the ground electronic state of thesecondary radiating matter and one that is not fully or completelyabsorbed by the intervening space between the source of primaryradiation and the source of secondary radiation. The purpose of same isto introduce energy into rotational and vibrational modes into theground electronic state of the source of secondary radiation and at thesite of secondary radiation generation, although it is not a requirementof the instant invention that only this site is populated with energy.By populating these vibrational bands through thermal radiation, we areessentially freezing out these modes for deactivation of the excitedstates of the secondary emitter. By populating these modes with infraredradiation, it is the same as if we sterically prevent the vibrationsfrom take place in the first place: an effect known to dramaticallyincrease the fluorescence Quantum Yield.

As by way of traditional modern molecular photochemistry, thefluorescence Quantum Yields of luminescent aromatic compounds arenormally quite high as they are rigid molecules, both in the groundstate and in the excited state from which they luminesce^(⋅) (Turro,Ramamurthy, & Scaiano, 1965) Rigidity can prevent cis: transisomerization, a competing mechanism of deactivation from excitedelectronic states. (Xu, et al., 2016)

The IR spectroscopic peaks observed in the ground electronic state ofCe³⁺:YAG have been attributed to the vibrational transitions as shown inTable 5 which is referred to as “Vibrational energies of YAG crystals”.Recall, as we previously noted, another referenced work indicated FT-IRspectroscopic peaks at around 790 cm⁻¹ which equates to 12.66μ and 2.26kcal/mol and another spectroscopic peaks around 3432 cm⁻¹ which equatesto 2.9μ and 9.8 kcal/mol. (He, et al., 2016)

An example of a source of secondary radiation is a Ce³⁺:Nd³⁺:YAGnanoparticle as elsewhere described (Wang, et al., 2015) or matter withthe chemical composition of Ce³⁺:YAG or a wide variety of inorganic andorganic luminophores with radiative lifetimes of less than 100nanoseconds at their localized temperature as defined by theKennard-Stepanov equation. For those sources of secondary radiation thatare sensitive to higher localized temperatures, they are encapsulated ina protective environment which may include exposure to a non-cooling gassuch as xenon, krypton or argon or in the alternative, a reducing gassuch a hydrogen that is diluted with a non-cooling gas such that thenon-cooling gas makes up at least 50% of the composition of the gas.

Incidentally we have noted that the orthodox definition of luminescenceis electromagnetic radiation with a chemical potential, μ. In contrast,thermal radiation is electromagnetic waves or photons with zero chemicalpotential. Conventional treatments of the difference presume asignificant magnitude to the chemical potential, evidenced by a sharpcut-off in the excitation spectrum for a luminescent body and theinference that thermal radiation and luminescence (cold radiation) havedistinctly different properties. However, as the temperature of a heatbath, at which a luminescent body absorbs incident radiation, increases,the chemical potential of the luminescent state decreases, and thesubsequent luminescence becomes more thermal radiation-like. This is aninevitable consequence of the luminescent excited state looking morelike the ground state, at higher temperatures, from which the initialexcitation takes place, as the chemical potential decreases from, forexample, 3.0 eV (69 kcal/mol) to 0.3 eV (6.9 kcal/mol; 2,420 cm⁻¹; 4,132nm) to 0.03 eV (0.69 kcal/mol; equivalent Temperature 348 K; Boltzmannpopulation (at 298 K) 31%) to 0.003 eV (0.069 kcal/mol) to zero eV.

We return to the reference of Rotschild wherein it is stated:

“By definition, the regime where photoluminescence and thermalexcitations compete for dominance is when μ approaches zero. At μ=0, theradiation is reduced to the thermal emission rate.” (Rotschild, 2017)

There is no abrupt cut-off between photoluminescence and thermalradiation: we emphasize the competition occurs as the chemical potentialapproaches, but is not yet, exactly zero. Indeed, since the chemicalpotential is defined as zero for those systems with a perfectequilibrium between the radiation field and matter, an idealized systemthat is never truly realized, it is clear then that those systemsgenerating photoluminescence, governed by the rules of thermodynamicsand quantum mechanics, will always concurrently yield thermal radiation,and the former will look more like the latter as the energy gap betweenelectronic states decreases.

The thermodynamic treatment of radiation as claimed herein relies on thechemical and thermal equilibrium among the electrons in the excitedstate atomic or molecular orbitals and among the electrons in the groundstate orbitals. If these equilibria exist, as the instant inventionclaims, the emission of luminescent radiation is an equilibrium problemas much as the emission of thermal radiation. That is to say that thesteady state arrival of electro-magnetic radiation is the same as theheat bath in which the matter resides and is stimulated either byphotons or phonons, whatever the case may be, respectively.

Entropy of Mixing: Its Origin. Entropy of mixing is a thermodynamicreality normally discussed and associated with the examination of idealgases. We are reminded that in the early models for thermal radiationthe parallels with ideal gases were observed, contemplated, and led to asearch for a particle nature of radiation. The well-known Gibbs paradox(Jaynes, The Gibbs Paradox., 1992) is associated with a thoughtexperiment whereby a chamber with a partition has the same ideal gas (insub-chambers) on both sides of the partition. With removal of thepartition, there is no entropy of mixing as the identity of the gas fromeither chamber cannot be distinguished among themselves, as opposed towhen each chamber contains a different gas, even if only by an atomicisotope. The key references Mr. Jaynes offers of Prof. Gibbs are:

1) Gibbs, J. Willard (1875-78) “On the Equilibrium of HeterogeneousSubstances”, Connecticut Acad. Sci. Reprinted in The Scientific Papersof J. Willard Gibbs, by Dover Publications, Inc., New York (1961);

2) Gibbs, J. Willard (1902), “Elementary Principles in StatisticalMechanics”, Yale University Press, New Haven, Conn. Reprinted in TheCollected Works of J. Wil lard Gibbs, Vol. 2 by Dover Publications,Inc., New York (1960).

[When it is said that two identical gases mix without change in entropy,it does not mean that the process can be reversed without change inentropy. Rather it means that we can return to a thermodynamic statethat is indistinguishable from the original one with respect tomacroscopic properties. Similarly, when we say that there is an increasein entropy when two distinguishable gases mix, it does mean that thereis a decrease in entropy when a process allows for return to athermodynamic state that is indistinguishable from the original one withrespect to macroscopic properties.] (Jaynes, The Gibbs Paradox., 1992)While this may seem unusual, one thing that is clear is that the entropyof mixing for indistinguishable ideal gases is completely independent ofthe nature of the gases. Further, Gibbs concluded that it is not anymore impossible to return two different molecules to a thermodynamicstate indistinguishable from its original state than it is to return twoidentical gas molecules, once mixed, to a thermodynamic stateindistinguishable from its original state. While one cannot affect thisseparation without impacting any variables of the thermodynamicstate—thus the entropy must change in de-mixing—it has been noted thatit might happen at some later point in time without any intervention,thus impossible becomes improbable.

In contrast to Gibbs Paradox for ideal gases (if one had even existed inthe view of Gibbs), it has been shown that if each of two cubic chamberscontain the same (indistinguishable) thermal radiation, removal of thepartition (increasing the volume) leads to an identifiable entropy ofmixing under isothermal conditions: (Jaynes, The Gibbs Paradox., 1992)(Sokolsky & Gorlach, 2014)

A. “When the partitions between cubic cavities are removed, neweigenfrequencies appear for the electromagnetic field inside. If thetemperature of the system had been kept constant, then the statisticalweight of a given system state has increased (when new modes, appear thenumber of ways to distribute the energy between these modes increases).Thus, the entropy has increased.” The entropy of mixing is in fact thesource of information entropy.

B. The entropy increase from the mixing of radiation is due to theincrease in radiation modes (photon numbers) offset in part by adecrease in temperature post-mixing: “it is shown that if the partitionbetween two adjacent identical cuboid cavities with the photon gases atthe same temperatures and pressures is adiabatically removed, thetemperature of radiation decreases in the obtained composite cavitythough the total number of photons increases”. [This distinguishes anideal gas from what may be called a photon gas—the lack of conservationof numbers in the latter case, which shall be more fully commented onhereinafter.] This is a recurrent theme in the context of maximumentropy and the return of radiation to an equilibrium state as aconsequence of a perturbation to the equilibrium (Le Chatelier'sprinciple—if a system at equilibrium is disturbed by a change intemperature, pressure, volume, or the concentration of one of thecomponents, the system will shift its equilibrium position so as tocounteract the effect of the disturbance).

C. In Down-Conversion, the entropy of mixing is of two radiation sourcesthat are unequal and distinguishable by virtue of their spectrum, (e.g.;blue radiation that is scattered and yellow radiation that is producedin any direction) among other parameters.

For example, an entropy of mixing term comes out of a derivation by Knoxfor the term

$\frac{\partial S}{\partial N}$

(change in entropy to the radiation field with addition of one (emissioninto) or elimination of one (absorption out of) photon) where S isentropy and N is the number of photons of frequency v present. (Knox,Thermodynamic and the Primary Processes of Photosynthesis., 1969)

There are other causes of energy-wasting (i.e.; work-wasting) in lightemitting diodes using Down-Conversion that are not related to theentropy of mixing. Consequently, despite the energy preservingcharacteristics of light emitting diodes when compared to other types ofGeneral Illumination lamps, there remains many cumulative energy lossesand the observable overall generation of heat is often explained asbeing a consequence of these losses. (Weisbuch, 2018) (Schubert, 2018)

It has been previously noted in the instant invention that many haveexpressed that one source of the energy inefficiency in lamps containingluminophores and light emitting diodes, is the Down-Conversion ofprimary radiation to secondary radiation. The latter (secondaryradiation) is of a lower internal energy than the former (primaryradiation) and, considering the conservation of energy, the differenceis said to be dissipated in the form of heat proximate to the body whereDown-Conversion takes place. This is an important consideration in thatif this loss is, in fact, dissipated in this manner, positioning of thephosphor remote to the light emitting diode die would do little toprevent heat from being produced at the secondary source of radiation,if preventing, rather than mitigating, mediating or ameliorating theeffects of heat is indeed the claim. (United States of America PatentNo. 2012/0280256, 2012) Indeed, there seems to be little improvement inluminance if remote phosphors experience the same temperatures (^(˜)85°C.) as when the phosphor is in contact with the light-emitting diodedie.

While the loss of efficiency due to the energy difference inDown-Conversion between the states which are populated after absorptionand of those states occupied from which emission takes place is nearlyuniversally quoted, there is little experimental evidence to supportthis description, the best proof of which would be based on calorimetry.Indeed, interpretations published by Mr. Knox on Kennard-Stepanov“parametric temperatures” is important to study: “(In a selection of)cases in which the anticipated linear relation is found . . .‘incorrect’ temperatures are implied by the data. The first reaction tosuch a finding may be to envision some heating of the local environmentby excess photon energy, and some attempts were made to connect thiswith Jab

oński's ideas about excess excitation energy dissipation.

“This point of view does not withstand close scrutiny in the KS-relationcontext . . . A competing hypothesis to explain the elevated T* is basedon inhomogeneous broadening.” (Knox, Excited-State Equilibration and theFluorescence-Absorption Ratio., 1999) The presumption of excess energyand its deleterious effects on subsequent processes appears not to besubstantiated to the point that the widely believed presumption itselfmay not be considered as unambiguously proven (although it has been anappealing explanation associated with the quantum mechanical descriptionof luminescence). (Van Der Hofstad, Nuzzo, Van Den Berg, Janssen, &Mekers, 2012) Inhomogeneous broadening is a consequence of the sourceradiation experiencing multiple different environments all of whichimpact, differently, the vibrational levels of the ground and excitedelectronic states.

Another alternative interpretation of line broadening is due to theincrease in entropy of the radiation field as a consequence of theconjugate variables, the energy and time uncertainty principles: “abasic assumption implicit in the application of thermodynamics to theelectromagnetic field is that the laws of thermodynamics are locallyvalid for radiative emission and absorption processes. This means that acertain minimum amount of entropy must be created by the radiativeprocess itself . . . by considering the extreme case in which thespontaneous emission of a natural spectral line is the only processtaking place, (that) this assumption is correct, and that its validityis essentially a consequence of the uncertainty principle as expressedby the reciprocal relationship between natural line breadth andlifetime.” The significance of this reasoning is that the entropy whichis a consequence of a Stokes shift is distributed to the radiation fieldand not to the surrounding energy bath, as is normally and historicallyassumed to be the case (i.e.; Jab

oński's ideas). (Weinstein, 1960)

Admittedly, the measurement of energy of an emission which impacts themeasurement of the time, and the non-commutative analogue, for whichemission takes place, is only problematic when the rates of emission arevery fast, as Planck's constant, h/2π, is an extremely small number.

Down-Conversion of course, by definition must utilize the hereinbeforereferenced Stokes fluorescence (usually; but in certain cases, also,phosphorescence) shift at the source of secondary radiation. (Stokes,1852) An often cited model for fluorescence generally (but not always)proceeds through three successive steps, i.e.; (1) absorption ofradiation to reach non-stabilized excited states (Franck-Condon states),(2) stabilization of the excited state by vibrational relaxation (i.e.;Jab

oński's ideas) and internal conversion (IC) to populate the lowestexcited state (Kasha State), and (3) radiative decay resulting in theemission of a photon. (Birge, 1926) (Condon, 1926) (Franck & Dymond,1926) (Kasha, 1950) (Galas, et al., 2018) Even absent the Stokes shiftrequirement (within Down-Conversion) for luminescence (a photochemicalor photophysical process), or Kasha's rule, not all energy incident on aluminophore can be exploited as useful energy (i.e.; to perform work).This is a function of incident intensity and temperature: the higher thetemperature and the lower the intensity, the less useful work that canbe performed. In the end, this is a matter of efficiency and notcontrary to the statement made by Dr. Einstein whereby he states withrespect to the Stokes shift and luminescence that “the intensity oflight produced must-other things being equal-be proportional to theincident light intensity for weak illumination, as every initial quantumwill cause one elementary process of the kind indicated above,independent of the action of the other incident energy quanta.Especially, there will be no lower limit for the intensity of theincident light below which the light would be unable to producephotoluminescence.” (Einstein, Concerning an Heuristic Point of ViewToward the Emission and Transformation of Light., 1905)

The best evidence for the energy difference in the Stokes shift beingdissipated in the form of heat to the local environment would be in theuse of calorimetry. In recent decades, a new technique calledPhoto-Acoustic calorimetry (PAC) provides for time-resolved examinationof comparative heat flows. As noted by experts in this field “a (A)chromophore absorbing a short laser pulse releases a heat pulse whenthat chromophore undergoes radiationless transitions. This heat pulseproduces a thermoelastic expansion of the medium that, if confined intime and space, launches a photoacoustic wave. Time-resolvedphotoacoustic calorimetry (PAC) is based on the detection ofphotoacoustic waves using ultrasonic transducers. PAC is often thetechnique of choice to measure transient volume/enthalpy changes ofshort-lived intermediates, and their corresponding lifetimes, due toradiationless processes subsequent to electronic excitation by a shortlaser pulse.” (Schaberle, Rego Filho, Reis, & Arnaut, 2016)

Jab

oński's ideas or Kasha's rule is based on a very fast radiationlessdecay to the lowest vibrational energy of the excited singlet stateafter which there is normally a relatively longer time period before theStokes fluorescence occurs. Of course, for Down-Conversion as practicedfor General Illumination, generally the chromophore, the phosphor, isnot in a medium that is a liquid to which the PAC is normallyapplicable. Thus, it is not generally clear that PAC can be used toascertain whether the entropy generated from the radiationless decayfrom the lowest vibrational level of the electronic excited state fromwhich luminescence takes place.

More importantly, the technique, PAC, has been used to measure theenergy of the lowest vibrational level of the electronic excited stateand the radiative lifetime associated therewith and that the techniquegives results near that of spectroscopic examinations. Most importantly,PAC has not been used to demonstrate that the Stokes' shift itself, thatvery fast process by which the secondary emitter loses energy from ahigher vibrational level of an electronic excited state to the lowestvibrational level of the electronic excited state, raises the localtemperature in the medium in which the secondary emitter is enveloped.Rather the technique has been used to measure the radiative lifetimeafter the Stokes' shift has already taken place. Finally, even absentall of the above limitations, the technique is not one that quantifiesthe amount of entropy that is dissipated through conduction versus thatwhich is dissipated through luminescent or thermal radiation, in otherwords, the key “proportional” characterization as articulated by Dr.Einstein.

Excited State Thermodynamics. The thermodynamics of photochemistry andphoto-physics is generally not considered (as is the thermodynamics ofthe incident radiation itself) in modern research on excited electronicstates. (Ross R. T., 1967) (Case & Parson, 1971) (Saltiel, Curtis,Metts, Winterle, & Wrighton, 1970) (Berthelot & Borns, 1926) This is thecase even though the thermodynamics of luminescence, for example, has along history of scientific pursuit and is often said to be rediscoveredmany times over the past century and one-half. The two areas ofcontemporary scientific research that does seek to incorporatethermodynamics of radiation is photosynthesis and solar cells. In bothcases the source of primary radiation is itself thermal radiation sothat a thermodynamic consideration is perhaps more natural to invoke.However, there is no fundamental reason that thermodynamics ofphoto-initiated processes needs to be restricted to primary radiationthat is thermal radiation.

As Ross & Calvin noted, losses in a photo-initiated process needs toconsider “the entropy associated with the absorbed radiation; in otherwords, free energy is not the same as energy.” Secondly, with respect toenergy storage, which was their consideration, “If an absorber were inequilibrium with a radiation field, then it would reradiate at the samerate at which it received photons, meaning that the Quantum Yield forenergy storage processes would be zero. In order to get a net retentionof photons, the entropy of the absorber must be greater than the entropyof the radiation field”. (Ross & Calvin, 1967) This view of Ross andCalvin can be used to support the Mauzerall Principle that aphoto-process under steady state of illumination is not an equilibriumprocess as there must be a net retention of photons for an excited stateto be available in which to retain energy. (Kahn, 1961)

The instant invention is principally based on a new understanding inDown-Conversion of the entropy of radiation and entropy can be evaluatedfrom a statistical mechanics perspective irrespective of equilibriumthermodynamics. Thermodynamics is also envisioned using reversibleprocesses. As we shall hereinafter disclose, reversible processes revealthe maximum efficiency; for irreversible photo-processes, the actualefficiency is simply lower than the maximum efficiency but in analogythereto. Further, in the instant invention, we are not interested inenergy storage but energy dissipation with a steady state of emission ata rate slower than that of the initial excitation.

In part, the impact of thermodynamics (e.g.; the impact of entropy) isdependent on the temperature, T, in which the reactions are performed.Radiation-initiated processes are normally invoked as alternatives tothermal initiated processes. Hence, most photo-chemical andphoto-physical examinations takes place at ambient temperature or lesswhere the impact of the entropy term is less important. Even in thosecases where photoinitiated entropy-controlled chiral reactions have beenstudied, the entropy of the incident radiation is not considered andassumed to be lost (due to Kasha's Rule) to the environment (i.e.; Jab

oński's ideas) and play no further role. (Inoue, 2004) In other words,the enantio-specificity is a consequence only of matter and not theradiation field within which the matter is resident.

On the other hand, photochemistry with circularly polarized radiation toyield chiral matter is well known as is circularly polarizedluminescence from organic light emitting diodes with asymmetry. (Kagan,Moradpour, Nicoud, Balavoine, & Tsoucaris, 1971) (Kuhn & Knopf, 1930)(Farshchi, Ramsteiner, Herfort, Tahraoui, & Grahn, 2011) (Heyn, 2011)[Thermal radiation generated in a magnetic field is circularlypolarized; magnetic field-free circularly polarized thermal radiationcan also be produced, from matter that has no mirror symmetry.]Perceptively, it has been noted that an atom (with a ground electronicstate and an excited electronic state) exposed to blackbody radiation isa thermodynamic system and only at absolute zero is the Helmholtz freeenergy and the energy of excitation equal so that all of the photonicenergy can be used to perform work. (Ford, Lewis, & O'Connell, 1987)Since blackbody radiation is such with a maximum entropy for thespectral distribution necessary to be absorbed by the ground statephoto-reactant, it is inevitable that at elevated temperatures of thephoto-reactant, the efficiency of the reaction is the lowest whenblackbody radiation, as opposed to other radiation sources, is theinitiator of the photoreaction, regardless of the reaction mechanism.

Historically, examination of photo-chemical processes has inherentlypresumed, by conspicuous omission, that the energy of excitation, E, isequal to the enthalpy, H, available for subsequent reaction, as it isassumed there is no PV work performed concurrent with excitation.[However, see report of volume contraction subsequent tophotoexcitation, usually due to cis-trans isomerization. The rate ofisomerization is so fast that it can be considered to occur with(concurrent with) photoexcitation.] (Mauzerall, Gunner, & Zhang, 1995)(Gensch & Viappiani, 2003) “In most optical systems P and V areconstant, thus the change in PV can be excluded from generating work. Anexception to this general rule is the phenomenon of sonoluminescence,where UV emission is generated as sound is converted into light througha drastic change in PV.” (Manor, Kruger, & Rotschild, Entropy drivenmulti-photon frequency up-conversion., 2013)

The thermodynamics of electromagnetic radiation is core to the instantinvention and shall hereinafter be more completely revealed. Therelationship between the energy of excitation and the free energyavailable to perform work is often assumed to not be impacted byentropy. The thermodynamic equation for a photo-initiated process is:

dG=dH−TdS=dE+d(PV)−TdS=hv ₀ +PdV−TdS  (16):

The enthalpy is the change in energy concurrent with excitation plus anyPV work so performed; assuming no PV work concurrent with excitation(within the period of time that an electromagnetic wave passes, at thespeed of light, by a chromophore of a particular molecular size), thenthe enthalpy change is that of the photonic excitation as commented on,more eloquently, by others:

“If excitation does not change the pigment's volume significantly, theincrease in enthalpy when the pigment absorbs a photon is the same asthe change in internal energy (hv₀). The pigment's increase in Gibbsfree energy per photon absorbed (ΔG_(p)=hv₀−TΔS) then is zero at lowlight intensities where the enthalpic and entropic terms cancel andincreases to hv₀ at high intensities as

$\left( \frac{n_{e}}{n_{g}} \right)$

approaches 1 and the entropic term drops out.” (Knox & Parson, Entropyproduction and the Second Law in photosynthesis., 2007) (Yarman,Kholmetskii, Arik, & Yarman, 2018)

This rather interesting insight attributed to Mr(s). Knox and Parsons,by way of the references included herein, is that the entropy ofexcitation is essentially the same as our entropy of mixing (in thiscase, at the site of our source of “primary radiation”) at lowintensities however, the Boltzmann thermal distribution governs andTΔS=ΔH=hv₀. That is, there is no free energy available to perform workwith very little intensity. At high intensities n_(e)=n_(g) and

${k_{B}\mspace{14mu}{\ln\left( \frac{n_{g}}{n_{e}} \right)}} = {{k_{B}{\ln(1)}} = 0}$

so that ΔG_(p)=hv₀−TΔS=hv₀−T(0)=hv₀. That is at high intensities, thefree energy to perform work with each new photon absorbed is essentiallythe photonic energy of excitation. Of course, the system has to bedesigned so that the work that is performed is the desired work and notsome other means of matter so excited returning to the ground electronicstate. This thermodynamic argumentation is important in that it shouldremind the casual observer that the utility of incident radiative energyis different at very high intensities than it is at very lowintensities.

The lucid and convincing argumentation of Mr. Knox and Mr. Parsons hasbeen similarly discussed and debated elsewhere between one Mr. Duysensand one Mr. Kahn^(:) (Kahn, 1961)

“A few years ago, I gave a method, mainly based on the second law ofthermodynamics, for calculating the maximum efficiency with which lightenergy can be converted into so-called high-grade energy such as work orGibbs free energy. (Duysens, The Path of Light Energy inPhotosynthesis., 1959) (Duysens & Amesz, Quantum Requirement forPhosphopyridine Nucleotide Reduction in Photosynthesis., 1959) Thisefficiency approached zero with decreasing light intensity and withincreasing temperature of the system for conversion of light energy. Anessential condition is that the light is diffuse (so that its intensitycannot be enhanced by means of a lens), or that the light-convertingdevice is as efficient in diffuse as in non-diffuse light. Thefollowing, more interesting, reasoning by Dr. Kahn is based upon thestatements, with which I agree: (A) the kinetic energy with which anelectron is liberated from a photo-sensitive surface is independent oflight intensity, even if this intensity approaches zero; (B) thephotocurrent is proportional to light intensity. The author (Dr. Kahn)concludes, but incorrectly, as I hope to demonstrate, that themacroscopic work done per unit of absorbed light energy is independentof intensity, and remains a constant positive value, even if the lightintensity approaches zero. This conclusion contradicts my result thatthe efficiency of energy-converting devices must approach zero, if theintensity of the light approaches zero.” (Duysens, A Note on Efficiencyfor Conversion of Light Energy., 1962) For reversible processes, themaximum efficiency approaches zero at low intensity and or hightemperature; for irreversible photo-processes the actual efficiency islower than the maximum efficiency.

The essential point of Mr. Duysens is to remind that for aphotoreaction, the maximum efficiency approaches zero at low intensityand or high temperature. We also remind, conversely, that at highradiation intensities a photo-reaction's efficiency may reach a maximumvalue: singlet-singlet annihilation, which is detrimental to operationof high brightness light-emitting devices, occurs at high radiationintensities.

Of course, the preceding discussion by and between Mr. Duysens and Mr.Kahn is reminiscent of earlier debates of the photoelectric effect firstexplained by Dr. Einstein in support of his premise that radiation had adistinctly particle-like behaviour. The clear thermodynamicargumentation is that the free energy available for work from aphoto-initiated process approaches zero with decreasing light intensityand with increasing temperature of the system. Mr. Duysens isdistinguishing between macroscopic work that can be performed (as afunction of radiation intensity) versus the kinetic energy generated andthe photocurrent produced, which are independent of and proportional tothe intensity of the incident radiation, respectively. Dr. Einstein'streatment included comments that the kinetic energy with which anelectron is liberated from a photo-sensitive surface is independent ofthe light intensity (even if the incident light intensity is near zero;see also Dirac). Those dismissing the thermodynamic treatment presumethat there is a relationship between the magnitude of the “kineticenergy produced” and the efficiency of the work performed, even thoughit is well known that the efficiency of a photo-initiated process (theQuantum Yield) is rarely unity. More specifically, Dr. Kahn limitedthermodynamic treatment as not in violation of the basic principles ofquantum mechanics to this particular scenario:

“If one were to construct a system such that the photon source and thephotoelectric specimen were placed in a perfectly reflecting envelopeand sufficient time had passed for the various modes of electronexcitation and decay in the specimen to be in equilibrium with thephoton gas in the envelope, then the laws of thermodynamics could beapplied to this system of photoelectric specimen and light source.”(Kahn, 1961)

We infer that since a perfectly reflecting cavity with only a photon gaswould never reach equilibrium, that then it is the photoelectricspecimen that must be perfectly absorbing and perfectly emitting so thata blackbody radiator would reach equilibrium.

As we shall hereinafter define, once the initial excited state ispopulated, the subsequent processes may also be affected by entropic andenthalpic parameters that may impact the free energy available toperform subsequent work.

It shall be emphasized in the instant invention that the efficiency of aset of photoreactions approaches zero not only as the intensity of theincident radiation approaches zero but also as the temperature reaches amaximum. The inefficiencies associated with increasing temperature areinevitable and cannot be avoided, regardless of the mechanism by whichentropy is increased in the underlying photo-processes. This is not tosay that in certain cases increasing temperature cannot be used to availa system to a particular temperature-enhanced photo-process. It simplymeans that at some point, in general, with increasing temperature allpossible spontaneous photoreaction pathways will be a means ofincreasing total system-wide inefficiency. The utility and novelty ofthe instant invention therefore is that increasing temperature, up untila certain point, is utilized to overall increase efficiency ofDown-Conversion by populating certain vibrational modes using thermalradiation or infra-red spectral radiation from a source that may bedifferent than the source of primary radiation.

The excitation of a chromophore is defined herein to take place at nomore than 1 atto-seconds (1×10⁻¹⁸ seconds), whereas subsequent processessaid to be photo-initiated can take place in 1) femto-seconds (10⁻¹⁵) or2) pico-seconds (10⁻¹²) or 3) nano-seconds (10⁻⁹) or 4) micro-seconds(10⁻⁶); or 5) milli-seconds (10⁻³), 6) seconds or 7) hours. For example,it is reported that it takes between five (5) and fifteen (15)atto-seconds to ionize an electron from an atom after theelectromagnetic wave first encounters the atom. (Ossiander, et al.,2017) Luminescence after photo-excitation in Down-Conversion takes placeoptimally between 1 and 100 nanoseconds; that is to say that the optimalrate of luminescence is from 10⁷ to 10⁹ reciprocal seconds.

Radiometry. Before we go on, the discussion that follows usesterminology that is best presently defined, for clarity. Thoseradiometric terms identified as “spectral” are those for which the unitof measurement include, for example, “per Hz⁻¹” if frequency is thespectrometric term used. When the spectrometric function is integratedout, there is a corresponding radiometric term that remains with “persteradians”. Generally, the instant invention is concerned with theradiation field emanating from an internal source, as opposed to thatfalling upon an external surface. The geometric model presumes a pointsource that can emit in all dimensions that make up a sphere. There aretwo additional terms: the area in which the radiation field is exitingand the angle from which it is radiating. The aforementioned steradianterm is actually a dimensionless derived SI unit as it is the area onthe surface of a sphere that is the square of the radius from the centerof the sphere into which the point source emits. The radiometricmeasurements that are calculated that include the angle of measurementinclude, by convention and for clarity, the steradian unit even thoughthe geometric concept is in reality unitless.

Hence, the integration of Spectral Radiance yields the Radiance;Spectral Intensity yields Radiant Intensity; Spectral Radiant Flux (alsocalled Spectral Radiant Power) yields Radiant Flux (also called RadiantPower). This is summarized in Table 1.

The difference between Radiance, Radiant Intensity and Radiant Flux haveeither the area or steradians integrated out. This summarized in Table2.

The spectral radiometric terms used herein are:

Spectral Radiance is Watts per steradian per square meter per hertz (SIsymbol L_(e,W,v)); Radiance of a surface per unit frequency (orwavelength, or wavenumber or angular frequency). The latter is commonlymeasured in W·sr⁻¹·m⁻²·Hz⁻¹. This is a directional quantity. Thismeasurement, Spectral Radiance, expressed in frequency, wavelengths orwavenumbers is one of the most common representation of thermalradiation. The thermal radiation model represented by Planck's Law is,in fact, a derivation of Spectral Radiance: L_(e,Ω,v) [W·sr⁻¹·m⁻²·Hz⁻¹].The Spectral Radiance is often provided a different symbol: B_(v) (ifevaluated in frequency). The Spectral Radiance is perhaps the mostcommon representation in Planck's Law (see Table 3).

Spectral Intensity is Watts per steradian per Hertz (SI symbolI_(e,Ω,v)) Radiant intensity per unit frequency or wavelength. ThusI_(e,Ω,v) [Watts sr⁻¹ Hz⁻¹] Since the frequency units remain, theSpectral Intensity has not been integrated over all frequencies, when sointegrated, it yields Radiant Intensity. Since Radiant Intensity has theunits Watts sr⁻¹, it has not been extended over the entire surface of asphere (in other words, it remains a directional quantity).

Spectral Flux is watt per Hertz (SI symbol ϕ_(e,v)); Radiant flux perunit frequency or wavelength. The latter is commonly measured in W·Hz⁻¹.Hence, ϕ_(e,v) [Watts Hz⁻¹]

The Spectral Exitance (or spectral emittance) is the radiant exitance ofa surface per unit frequency. Hence, M_(e,v) [Watts m⁻² Hz⁻¹]. It ismathematically assigned as

$M_{e,v} = {\frac{\partial M_{e}}{\partial v}.}$

The Spectral Exitance is one of the most common form of measurement forPlanck's Law (see Table 3).

The corresponding terms where the spectrometric function has beenintegrated out are:

Radiant Flux (also called Radiant Power) in Watts (SI symbol ϕ_(e)); ameasure of Radiant energy emitted, reflected, transmitted or received,per unit time. This is sometimes also called “radiant power”. Hence,ϕ_(e) [Watts].

Radiant Intensity in Watts per steradian (SI symbol I_(e,Ω)); a measureof Radiant flux emitted, reflected, transmitted or received, per unitsolid angle. This is a directional quantity. Hence, I_(e,Ω) [Watts sr⁻¹]Radiant intensity (often simply “intensity”) is the ratio of the radiantpower (also called Radiant flux) leaving a source to an element of solidangle d Ω propagated in the given direction. The term Intensity andBrightness is frequently misused in the field of radiometry soparticular attention to the formula or the units should be paid whenthese terms are invoked.

Radiance is in Watts per steradian per square meter (SI symbol L_(e,Ω)).Thus, L_(e,Ω) [Watts m⁻² sr⁻¹]. The Radiance is the Radiant Fluxemitted, reflected, transmitted or received by a surface, per unit solidangle per unit projected area. This is a directional quantity. Radianceis obtained by integrating the frequency, wavelength, or wavenumber ofSpectral Radiance over the entire region of frequency, wavelength orwavenumbers, respectively. The steradian units remain in the measurementof Radiance but the frequency units are integrated out. The presence ofthe steradian units indicate that the power per square meter has notbeen extended to all surfaces of a sphere from which the radiation emitsfrom or on to.

The Radiant Exitance (or the term radiant emittance) is the radiant fluxemitted by a surface per unit area. Radiant exitance is often called“intensity” in and this leads to considerable confusion. Hence,M_(e)[Watts m⁻²]. Parenthetically, it is noted that the radiant fluxreceived by a surface per unit area is called irradiance. The RadiantExitance is mathematically assigned as

$M_{e} = \frac{\partial\varnothing_{e}}{\partial A}$

and is the radiometric measurement in the Stefan-Boltzmann Law whereM_(e)=σT⁴ [Watts m⁻²] and where σ is the Stefan-Boltzmann constant.

There are two additionally common radiometric terms:

Radiant Energy in Joules (SI symbol Q_(e)); a measure of energy ofelectromagnetic radiation (sometimes denoted by E or U). Hence, Q_(e)[Joules].

Radiant Energy Density in Joules per cubic meter (SI symbol w_(e)); ameasure of Radiant Energy per unit volume. This measurement, oftencalled energy density and noted with the symbol μ_(e), was perhaps themost instrumental measurement when physicists were first able to providea model for thermal radiation that was in reasonable alignment betweenmodel and experimental data. The volume was calculable as the model wasbased on a known cavity design. Hence, w_(e)=μ_(e)[Joules m⁻³]. Theenergy density form of Planck's Law (in terms of frequency) is shown inTable 3.

Irradiance is Watt per square meter (SI symbol E). It is also calledRadiant Flux Density.

Radiometric terms are all functions of Energy; usually with respect tothe change in energy with time. There are no analogous radiometric termsfor entropy (see Table 4), other than the reference of Wright et. al.(Wright, Scott, Haddow, & Rosen, 2001) Hence, an effort to determine themaximum entropy per given energy is fraught with difficulty when theexperimental data is obtained using radiometry. Hence, when making suchcomparisons, it is most helpful to understand the units of measure forthe radiometric experiment so as to develop, by modelling, a comparableterm for the entropy of the radiation whose energy was measured.

By way of an example, keeping track of the proper units, we noted thatthe integration of the Spectral Radiant Exitance form of Planck's Lawyields the Radiant Exitance in terms of M_(e)=σT⁴ [Watts m⁻²]. TheSpectral Radiant Exitance is obtained by multiplying the SpectralRadiance by it steradians. It is noted that multiplying the entropyanalogue to the Spectral Radiance (for clarity, the Spectral energyRadiance) by it steradians yields the entropy analogue, the SpectralRadiant Entropy Exitance, and when subsequently integrated over allfrequencies, yields σT³, the entropy flow rate per unit area. [Watts K⁻¹m⁻²].

It is also helpful to define the term “power” and the term “work”. Poweris the amount of energy consumed per unit time; therefore, Radiant Fluxis sometimes called Radiant Power. Power, as a function of time, is therate at which work is done, so it can be expressed by this equation:P=dW/dT. Clearly, then, Work is a measurement of energy. Inthermodynamics, Work is differentiated from Heat in that the former isthe creation or consumption of energy that provides a useful function.Heat is the creation or consumption of energy that does not itselfprovide useful Work unless otherwise subsequently converted, always withless than unit efficiency, into Work by, for example, a heat engine. Arestatement is “if, while suffering a cyclic process, a body absorbsheat from its exterior, that body must also emit heat to its exteriorduring the process” or “In other words, the heat supplied to the bodyundergoing the cyclic process cannot be converted entirely into work:there must be emission of heat from the body as well. The cyclic processcannot operate with perfect efficiency.” (Evans, 2014)

As stated hereinbefore, reiterated for the purpose of summarization, 1)all matter generates and emits electromagnetic waves (i.e.; radiation,or light) based on the temperature of the matter; 2) this is calledthermal radiation which is an important means by which matter dissipatestransfers heat; and 3) other means of heat transfer are called heatconvection and heat conduction. (Cuevas & Garcia-Vadal, 2018) Althoughheat is energy and is measured in units of energy, heat in a physicalprocess is often associated with inefficiencies. Of course, heatinefficiencies can be partially reconverted into useful energy in theform of a heat engine. Heat not reconverted raises the temperature of anoperating system. Carnot efficiencies can be calculated for heat enginesand this is perhaps the principle reason that heretofore thethermodynamics of radiation has focused more when the primary radiationis thermal (its temperature is known) versus from luminescence.

There are many issued patents with respect to light emitting diode lampsthat claim to transfer unwanted heat by heat convection or heatconduction or both. (U.S. Pat. No. 10,107,487, 2018) Notwithstanding,there are some beneficial aspects of thermal radiation that make it moredesirable as an alternative means to dissipate heat. We recognize aso-called 4/3 factor (33% increase), a thermodynamic factor, that weshall hereinafter exploit. (Wright, 2007) (Chukova, 1976) Anotherelement of thermal radiation electro-magnetic waves that we shallexploit is that it is the only known “particle” where the conservationof numbers does not apply. (Buoncristiani, Byvik, & Smith, 1981)(Smestad, Ries, Winston, & Yablonovitch, 1990) (Meyer & Markvart, 2009)[Volume fluctuations is perhaps the only time that photon numbers do notchange as a response to changes in the equilibrium state.] (Norton,2006)

The luminescent state can be activated thermally so that its chemicalpotential is zero. “Excited states that produce visible radiation can beprepared thermally, but this happens at a very small rate. Indeed, sincewe expect the upward transition rate to be

$e^{- \frac{\Delta\; E}{k_{B}T}}$

times the radiative rate in fluorescence, we shall wait, on average,1.9×10¹⁴ years to perceive a red (ΔE=1.8 eV) thermal photon at roomtemperature.” This is a consequence of the exceptionally smallhigh-energy tail (the Wien limit) of the Planck distribution at T=295 K.(Knox, Thermodynamic and the Primary Processes of Photosynthesis., 1969)(Knox & Parson, Entropy production and the Second Law inphotosynthesis., 2007) (Knox, Excited-State Equilibration and theFluorescence-Absorption Ratio., 1999)

When a fluorescent state is excited directly, this tail is effectivelyamplified. The formalism of Kennard and Stepanov is essentially basedupon a shift of the zero of energy in the Boltzmann factor from theground electronic state to some principal excited electronic state(called standard fluorescence). (Gilmore, 1992) After the preparation ofthat state the system is assumed quickly to equilibrate with its thermalbath, sufficiently quickly that all of the observed emission originatesfrom the so-equilibrated states.” (Knox, Excited-State Equilibration andthe Fluorescence-Absorption Ratio., 1999)“However, it should not beassumed that a Boltzmann distribution of the excited electronicsublevels is identical to the Boltzmann distribution of the groundelectronic sublevels. Hereinafter, such an assumption will be referredto as ‘standard Fluorescence’.” (Gilmore, 1992)

Without further clarification at this point we simply refer to thefollowing: in general, “the number of photons in a photon gas is notconserved. However, light emission and absorption processes involveinteractions of photons with other quasiparticles, such as electrons,plasmons, excitons, polaritons, etc., and these interactions obey theconservation laws for energy, momentum, and angular momentum. Thus, thenumber of photons created or annihilated during these interactionscannot always be unrestricted. As a result, photons may derive not onlytheir temperature, but also chemical potential from these interactions.”(Boriskina, et al., 2016) (A photon gas is a thermodynamic model,paralleling that of an ideal gas, of electromagnetic radiation in acavity)

A unique element of thermal radiation, heat transfer from a material toa recipient through radiation, is that it is independent of the emittingmaterial's composition and the molecular bonding chemistry (that on anatomic scale defines the material's properties on a macro scale).Whereas we mentioned that there are three types of heat transfer, thereare really only two: thermal radiation, which is independent of thematerial's composition, and diffusion, which is transfer throughmolecular interactions. The latter is a function of, for example,molecular heat capacity; the former is generally thought to beindependent of molecular oscillations unique to and characteristic of aradiating material (e.g.; vibrations; rotation).

Thermal radiation is also different from thermal diffusion (conductionand convection) in that the latter depends on the temperature gradientsand physical properties in the immediate vicinity of the element that istransferring heat to another element of space. In thermal radiation,energy is transferred between separated elements (two separatedmaterials) without the need of a medium between the elements of space.(Siegel & Howell, 1972) A system can and usually does transfer heat byboth mechanisms (radiation and molecular diffusion) at the same time andthe total heat transferred is a sum of the magnitude for each process.The distance between the source of heat and the recipient of heat is animportant consideration, as well, in the context of thermal radiationand molecular or atomic diffusion. Indeed, we can look at three domainsof distance: far-field, near-field and contact. Thermal radiation in thefar-field is of course that represented by the sun and the earth.

The near-field is represented by the distance between source andrecipient being closer than the dominant wavelength of radiation thatdefines the thermal radiation which at ambient is near 10 μm and theheat flux transferred between two blackbody radiators is enhanced bymany orders of magnitude called the “super-Planckian” effect. This isattributed to the additional contribution of evanescent waves.

The heat flux enhancement in the near-field regime has good agreementwith theoretical models of macroscopic heat transfer, which, however,cannot fully describe heat exchange at distances down to a fewnanometres. In particular, such theories generally do not account forthe cross-over from near field to contact, in which case the objects areseparated by atomic distances and heat flux is mediated by conductivetransfer. (Kloppstech, et al., 2017) In the present invention, thetransfer of heat is not required to be in the range of atomic distances,that is to say, not required to be, and we can exclude, in the rangedefined hereby as contact.

To further clarify, phonons (a quantum of energy or a quasiparticleassociated with atomics vibrations in a crystal lattice) require matterto exist and cannot propagate in bulk vacuum, although they cantransport across vacuum gaps a few angstroms wide. Vacuum phonontransport is a parallel heat-transfer process to near-field thermalradiation, although the length scales over which the two phenomenadominate are different. (Sellan, et al., 2012) Herein we define contactto include phonon transport across gaps a few angstroms wide.

In Down-Conversion, the primary radiation so produced can yieldsecondary radiation with up to, no less than, unit efficiency. The termunit efficiency in photo-processes refers to the number of exitingphotons subsequently performing useful work to the number of incidentphotons. On an energy basis, the inefficiencies in the creation of heatare either manifested by that which is disseminated by radiation,thermal radiation, and that which is disseminated via diffusion(convection or conduction). Depending on the origin of radiation, lightcan be either a worse or a better source of energy to extract work from,compared to heat conduction. (Boriskina, et al., 2016)

In this spirit, a fundamental difference between a thermodynamic view ofphoto-initiated processes and a quantum mechanical view is whether theunits of radiation so produced (in some manner) can subsequently performuseful macroscopic work or not (considered in thermodynamics but not inquantum mechanics which focuses on the microscopic atomic lengths).

The spectral distribution of thermal radiation is from the longwavelength “fringe of the ultraviolet, the visible light region whichextends from wavelengths of approximately 0.4 mm to 0.7 mm, and theinfrared region which extends from beyond the red end of the visiblespectrum to about 1000 mm”. (Siegel & Howell, 1972) More than a centuryago, a model for thermal radiation had been developed based on theobservation called “ultraviolet catastrophe” whereby an algorithm wasdeveloped consistent with thermal radiation not extending deep into theultraviolet region but starting at the “fringe of the ultraviolet” evenat very high temperatures, such as that of the Sun.

A Hohlraum Model. The physics of thermal radiation is modelled after avacuum cavity (a radiator) whose walls are black; they absorb and thensubsequently emit all incoming light. There is admittedly a time elementto absorbing and emitting subsequently all light which is normally notconsidered in the treatment of thermal radiation. The vacuum cavityradiator model does not require that all light be initially absorbed norat all times absorbed, only that when the system reaches equilibrium,then all light is absorbed and subsequently all is re-emitted. This isthe essence of the model that appears to be very predictive and thuswidely believed to be operative.

The characteristics of a vacuum cavity radiator depends on the surfaceof the material that comprises the wall of the cavity, an enclosure, andis dependent solely on whether the incident thermal radiation isabsorbed or not: it is not dependent on the specific material itself. Inthis sense we posit that the material indifference of thermal radiationis similar to the material indifference of the entropy of mixing of twodifferent gases (consider Gibbs' Paradox, noted as the entropy of mixingis independent of the materials' composition, only that they bedistinguishable). (Paglietti, 2012) Indeed, as Gibbs himself stated:“But if such considerations explain why the mixture of gas-masses of thesame kind stands on different footing from mixtures of gas-masses ofdifferent kinds, the fact is not less significant that the increase ofentropy due to mixture of gases of different kinds in such a case as wehave supposed, is independent of the nature of the gases.” (Gibbs, 1906)

In general, when radiation (thermal, or not) is incident on a surface,some of the radiation is reflected and some of it is absorbed as theradiation travels through the material. “A careful distinction must bemade between the ability of a material to let radiation pass through itssurface and its ability to internally absorb the radiation after it haspassed into the body”. (Siegel & Howell, 1972) As an example, a mirroredsurface may reflect most of the incident radiation, but that radiationthat does pass into the body of the material will be strongly absorbed.Thus, the mirrored material is said to have strong internal absorptioncapability but, nevertheless, most of the incident radiation isreflected.

A blackbody is defined as an ideal body—one does not really exist in itsfull form—that allows all incident radiation to pass through thesurface, whatever its geometry and topology, and into the body of thematerial, i.e.; no radiation is reflected at the surface, subsequent towhich, after passing through the surface, all radiation in the body ofthe material is absorbed. This is true for all wavelengths and for allangles of incidence. Hence, the blackbody is a perfect absorber ofincident radiation. It is also the perfect emitter: at equilibrium. Thisis the relevant definition that defines the conceptual model but as weshall see, except for perfectly reflecting surfaces (again one does notreally exist in its full form), a cavity radiator at equilibrium, onceit is reached, will analytically provide results predicted by theblackbody vacuum cavity radiator model.

The initial consideration of cavity radiation is that known asKirchhoff's Law of thermal radiation, dating from 1859-1860, that may bestated as follows (using the symbols used within the hereinafterreferenced review by Mr. Johnson): (Johnson, 2016)

“For an arbitrary body radiating and emitting thermal radiation, theratio E/A between the emissive spectral radiance, E, and thedimensionless absorptive ratio, A, is one and the same for all bodies ata given temperature. That ratio E/A is equal to the emissive spectralradiance I of a perfect black body, a universal function only ofwavelength and temperature’. This radiance, I, is often referred tosimply as black radiation.” (Johnson, 2016)

We again note here that the term “radiance, I” and “spectral radiance,E”, referenced by Johnson are in our definition, Radiance with the unitsWatts per steradian per square meter (SI symbol L_(e,W)) and SpectralRadiance with the units Joules per second per steradian per square meterper hertz (SI symbol L_(e,W,v)); respectively. Of only historicalimportance, some have argued that the same observations and conclusionsattributed to Mr. Kirchhoff were originally discovered by Mr. Stewart.(Stewart, 1859)

Of importance to today's current understanding of thermal radiation forthe transfer of heat, modelled as a blackbody hohlraum, meaning inGerman a cavity or empty space, both Dr. M. Planck and Dr. A. Einsteinmade many important contributions. There has been a plethora of reviewsof the historical significance of development and understanding of thephysics of blackbody radiators with emphasis on the contributions of Dr.Planck and Dr. Einstein, the thermal radiation produced therefrom beingdependent only on the temperature of hohlraum. (Wu & Liu, 2010) Thetreatment of Dr. Planck is a quintessential classical physics treatmentand the equations produced therein are classical thermodynamics,although the treatment invoked as an ancillary justification astatistical method of counting, perhaps an obtuse one, that appeared torequire oscillator energy states of integer number. (Boyer, 1984) Theepistle by Thomas S. Kuhn with its reference to quantum discontinuity isa particular interesting and thorough consideration of the manytreatments produced not only in a short period of time (1894-1912) butthe many changeable accounts produced by the main characters as well:Dr. Planck and Dr. Einstein. (Kuhn T. S., 1978) We include as areference for prior art to our instant invention the work of Thomas S.Kuhn and incorporate the references included therein as our ownreferences of the prior art. The quantum discontinuity of the excitationand emission of the matter itself, that comprises the hohlraum, alludedto by Dr. Planck and most certainly reinforced as early as 1903 when Dr.Einstein stated “During absorption and emission the energy of aresonator changes discontinuously by an integral multiple of

$\frac{R}{N}\left( {\beta\; v} \right)^{''}$

where “R is the gas constant, N Avogadro's number, β a constant”. (KuhnT. S., 1978)

Interestingly, Dr. Planck initially had based his proof on aconsideration of perfectly reflecting cavities, rather than blackbodysurfaces, but containing “an arbitrarily small quantity of matter”arriving at the same result that Kirchhoff had obtained for perfectlyabsorbing cavities. Dr. Planck had thereby demonstrated that allcavities either containing some small amount of arbitrary matter, orequivalently having walls completely comprised of arbitrary matter, mustalso contain black radiation when at thermal equilibrium. (Johnson,2016) Black radiation is that which cannot be observed as distinct fromits surroundings: radiation within, and viewed within, a blackbodycavity. When viewed outside the cavity, of course, the radiation isvisible.

Summarizing the main difference between blackbody cavity radiators andthose that are not, those with perfectly reflecting walls—if any wouldsuch exist—are those that never reach thermal equilibrium. This is animportant observation in that if a cavity reaches equilibrium and is nota perfect reflector (otherwise it would not reach equilibrium) and hassome absorption, it will closely approximate a blackbody radiator. Howlong it takes for equilibrium to be reached is another matter. In thereview authored by Mr. Johnson, there are many excellent explanationsthat convincingly demonstrate the universal nature of cavity radiatorsas long at the walls are not completely reflecting and at least a minoramount of black matter is present therein. (Johnson, 2016)

The classical physics of thermal radiation are more elegantly stated byothers:

“Planck maintains that” . . . the radiation of a medium completelyenclosed by absolutely reflecting walls is, when thermodynamicequilibrium has been established for all colours for which the mediumhas a finite coefficient of absorption, always the stable radiationcorresponding to the temperature of the medium such as is represented bythe emission of a black body”. Note that the quoted statement coversboth the situation where the object absorbs and emits over allfrequencies, and the situation where some frequencies are not absorbedor emitted at all.” (Johnson, 2016) Historians have noted that, indeed,Mr. B. Stewart himself much earlier “made the theoretical leap” that forcavities with perfectly reflecting walls “the sum of the radiated andreflected heat together became equal to the radiation of the lampblack”.

Some reviews have suggested that Dr. Planck also introduced the conceptof a white body. (Planck, The theory of heat radaition (Translation),1914) The definitions used by Dr. Planck are:

“When a smooth surface completely reflects all incident rays, as isapproximately the case with many metallic surfaces, it is termed“reflecting”.

When a rough surface scatters all incident rays completely and uniformlyin all directions, it is called “white”.

A rough surface having the property of completely transmitting theincident radiation is described as “black”.

Dr. Planck defines black materials as those with a rough surface whichdoes not reflect; all incident rays falling on a black material passthrough (are transmitted through) Planck's geometrical surface and aresubsequently absorbed at some depth in the interior of the blackbody. Norays are reflected from the body even if the material in which it ismade is a poor absorber. In other words, “A white body is one for whichall incident radiation is reflected uniformly in all directions, anidealization exactly opposite to that of the blackbody.” (Cheng, 2014)It has been recently demonstrated that a graphene nanostructure not onlyshows a low reflectance comparable to that of a carbon nanotube arraybut also shows an extremely high heat resistance at temperatures greaterthan 2500 K. The graphene nanostructure, which has an emissivity higherthan 0.99 over a wide range of wavelengths, behaves as a standardblackbody material. (Matsumoto, 2013) Graphene fiber has also beenfabricated. (Xu Z. G., 2015)

A contemporary of Dr. Planck was Prof. Wien who has briefly beenmentioned hereinbefore. His work will be mentioned in more detailhereinafter as it is generally agreed that Dr. Planck's work was animprovement on the earlier disclosures by Prof. Wien of his eponymousWien's approximation. A detailed review of “Radiation Theory and theQuantum Revolution” by Joseph Agassi reveals from Prof. Wien's 1911Nobel Prize winning speech his focus on white cavities to replace blackcavities. (Agassi, 1993) The story repeated by Agassi in his indomitablestyle was that the blackbody thermal radiation remains the same even ifthen placed in a white body. See page 96 and 97 in Agassi's review.

Light Rays, Light Scattering, Light Reflection. This our first mentionof rays of light. Of course, “light is electromagnetic wave phenomenondescribed in the form of two mutually coupled vector waves, anelectric-field wave and a magnetic-field wave. Nevertheless, it ispossible to describe many optical phenomena in the form of scalar wavetheory in which light is described by a single scalar wavefunction,called wave optics. When light propagates through and around objectswhose dimensions are much greater than the wavelength, the wave natureof light is not readily discerned, so that its behaviour can beadequately described by rays obeying a set of geometrical rules. Thismodel is called ray optics or geometrical optics (the limit of waveoptics when the wavelength is infinitesimally small) and is the simplesttheory of light (as compared with quantum optics)”. (Saleh & Teich,1991)

We have previously mentioned both reflection and scattering. They aregenerally used to mean the same thing: light not penetrating the body ofthe material and therefore not being absorbed nor subsequently emitted.These optical properties of “rays of light” are generally surfacephenomenon associated with solid state materials. In this instantinvention, we posit that cavities whose walls are white scatter in allpossible directions all incident light. The difference betweenreflection from a white “scattering” surface and reflection from a“mirroring” surface is Lambertian reflectance and specular reflectance,respectively. In specular reflectance, each incident ray is reflected atthe same angle to the surface normal as the incident ray, but on theopposing side of the surface normal in the plane formed by incident andreflected rays. In Lambertian reflectance, the surface's luminance isisotropic, and the luminous intensity obeys Lambert's cosine law. Weherein equate scattering with Lambertian reflectance without regard tomaterials composition and assuming the scattered rays are isoenergeticwith the incident rays. [Models for scattering also includes those whereenergy is lost and even where energy is gained, from the environment.]For the blackbody radiator, the internal black surfaces are ones whereincident rays initially “pass through” and subsequently are “absorbed”as opposed to simply “absorbed”.

While ray optics is a simplification of wave optics, itself asimplification of quantum optics, it is a remarkably useful classicalmechanics model that is consistent with many historical experimentalresults on the interaction of light with matter. Further, withconsideration of the Correspondence Principle—when the quantum numbers(energy) is large, the quantum mechanical system behaves according to(in correspondence with) the classical description—ray optics helps setup the classical thermodynamic treatment of thermal radiation.

Ray optics also brings into consideration the principle of least action.More specifically, in ray optics, Fermat's principle or the principle ofleast time, named after French mathematician Pierre de Fermat, is theprinciple that the path taken between two points by a ray of light isthe path that can be traversed in the least time. Fermat's principle isthe link between ray optics and wave optics.

Fluctuations from Equilibrium. Equilibrium states are core tounderstanding of cavity radiation. Thermal fluctuations are randomdeviations of a system from its average state, that occur in a system atequilibrium. All thermal fluctuations become larger and more frequent asthe temperature increases, and likewise they decrease as temperatureapproaches absolute zero. As one example, a closed system in thermalequilibrium, such as a molecular gas in an adiabatic enclosure, ischaracterized by statistical fluctuations in energy. “Let us consider athermodynamical (sic) system, which we will divide into two parts: onerelatively small, called from now on “system”, and a larger one, withthe name “reservoir”. Exchange of energy between these two parts willresult in fluctuations of energy of the smaller “system” near a certainequilibrium energy E. General thermodynamic laws give a well-knownequation attributable to Dr. Einstein:” (Kojevnikov, 2002)

$\begin{matrix}{\overset{\_}{\Delta\; E^{2}} = {\overset{\_}{\overset{\_}{E^{2}} - {2E\overset{\_}{E}} + {\overset{\_}{E}}^{2}} = {{\overset{\_}{E^{2}} - {\overset{\_}{E}}^{2}} = {{k_{B}T^{2}\frac{dE}{dT}} = {k_{B}T^{2}C_{V}}}}}} & (17)\end{matrix}$

where

ΔE=∈=E−Ē=E−<E>  (18):

and where

E² is the mean of the square of the equilibrium energy E; and

Ē² is the square of the mean equilibrium energy; and that

ΔE and E are redundant symbols as are Ē and <E>.

Blackbody thermal radiation is said to have the maximum entropy toensure that any deviations from maximum entropy result in a return tothe state of equilibrium. The root mean square of the energyfluctuation, ΔE_(rms), is then

$\begin{matrix}{{\Delta\; E_{rms}} = {\sqrt[2]{\overset{\_}{\Delta\; E^{2}}} = {\sqrt[2]{k_{B}T^{2}C_{V}}.}}} & (19)\end{matrix}$

Here, C_(V), is the heat capacity at constant volume which has thethermodynamic identity of

$C_{V} = {\left( \frac{\partial E}{\partial T} \right)_{V}.}$

The above equation is derived from determining the entropy fluctuation,ΔS, corresponding to an energy fluctuation, ∈=E−<E>=E−Ē, in thefollowing form: (Irons, 2004)

$\begin{matrix}{C_{V} = {\left( \frac{\epsilon^{2}}{2T^{2}} \right)\left\{ \left( \frac{1}{\left( \frac{\partial{< E >}}{\partial T} \right)_{V}} \right) \right\}}} & (20)\end{matrix}$

If one knew the equilibrium energy, E, for the system in contact with areservoir, one could simply insert and take the derivative with respectto T. Regarding fluctuations from equilibrium associated with radiation,one does have knowledge of the energy, E. As we shall hereinafterdemonstrate, the energy density of a blackbody system (an equilibriumbetween thermal radiation and matter) is known and is:

$\begin{matrix}{{\mu\left( {v,V} \right)} = {{Z_{V}U_{V}} = {\overset{\_}{E} = {< E \geq {\frac{8\pi\; v^{2}}{c^{3}}\left\lbrack {{{hv}\text{/}e^{\frac{hv}{kT}}} - 1} \right\rbrack}}}}} & (21)\end{matrix}$

Recall that the Radiant Energy Density in Joules per cubic meter (SIsymbol w_(e)) is a measure of Radiant Energy per unit volume. The energydensity form of Planck's Law—there are many forms of this law—isobtained by multiplying the energy by 4π/c′ (Rybicki & Lightman, 1979)(Ore, 1955)

The form used by Dr. Einstein was the energy density of the system (as afunction of frequency and Temperature) times the volume, V, of thesystem, to be integrated over all frequencies

$\begin{matrix}{E = {{{\frac{8\pi\; v^{2}}{c^{3}}\left\lbrack {{{hv}\text{/}e^{\frac{hv}{kT}}} - 1} \right\rbrack}{Vdv}} = {D\left\lbrack {{{hv}\text{/}e^{\frac{hv}{kT}}} - 1} \right\rbrack}}} & (22)\end{matrix}$

where D is a constant. Note that

$\begin{matrix}{{\frac{\overset{\_}{E}}{D} = {\frac{< E >}{D} = \left\lbrack {{{hv}\text{/}e^{\frac{hv}{kT}}} - 1} \right\rbrack}}{then}} & (23) \\{{{dE}\text{/}{dT}} = {\frac{hv}{k}D{\left\{ \frac{e^{\frac{hv}{k_{B}T}}}{{T^{2}\left( {e^{\frac{hv}{kT}} - 1} \right)}\left( {e^{\frac{hv}{kT}} - 1} \right)} \right\}.}}} & (24)\end{matrix}$

We have found the term

$\frac{dE}{dT}$

as a consequence and we recall that

$\begin{matrix}{\overset{\_}{\Delta\; E^{2}} = {\overset{\_}{\overset{\_}{E^{2}} - {2E\overset{\_}{E}} + {\overset{\_}{E}}^{2}} = {{\overset{\_}{E^{2}} - {\overset{\_}{E}}^{2}} = {{k_{B}T^{2}\frac{dE}{dT}} = {k_{B}T^{2}{C_{V}.\mspace{76mu}{Then}}}}}}} & (25) \\{\mspace{76mu}{{\overset{\_}{E^{2}} - {\overset{\_}{E}}^{2}} = {k_{B}T^{2}D\frac{hv}{k_{B}}\left\{ \frac{e^{\frac{hv}{k_{B}T}}}{{T^{2}\left( {e^{\frac{hv}{kT}} - 1} \right)}\left( {e^{\frac{hv}{kT}} - 1} \right)} \right\}}}} & (26) \\{{\overset{\_}{E^{2}} - {\overset{\_}{E}}^{2}} = {\left\{ {{Dhv}\left\{ \frac{e^{\frac{hv}{k_{B}T}}}{{T^{2}\left( {e^{\frac{hv}{kT}} - 1} \right)}\left( {e^{\frac{hv}{kT}} - 1} \right)} \right\}} \right\} = {\left\{ {{Dhv}\left\{ \frac{e^{\frac{hv}{k_{B}T}}}{{T^{2}\left( {e^{\frac{hv}{kT}} - 1} \right)}\left( {e^{\frac{hv}{kT}} - 1} \right)} \right\}} \right\} = {\left\{ {{Dhv}\left\{ \frac{e^{\frac{hv}{k_{B}T}}}{{T^{2}\left( {e^{\frac{hv}{kT}} - 1} \right)}\left( {e^{\frac{hv}{kT}} - 1} \right)} \right\}} \right\} = {\left\{ {{Dhv}\left\{ {\frac{1}{\left( {e^{\frac{hv}{kT}} - 1} \right)} + \frac{1}{\left( {e^{\frac{hv}{kT}} - 1} \right)\left( {e^{\frac{hv}{kT}} - 1} \right)}} \right\}} \right\} = {\left\{ {\frac{D}{\left( {e^{\frac{hv}{kT}} - 1} \right)}\left\{ {\frac{hv}{1} + \frac{hv}{\left( {e^{\frac{hv}{kT}} - 1} \right)}} \right\}} \right\} = {\left\{ {\frac{D}{\left( {e^{\frac{hv}{kT}} - 1} \right)}\left\{ {\frac{hv}{1} + \frac{hv}{\left( {e^{\frac{hv}{kT}} - 1} \right)}} \right\}} \right\} = {\left\{ {< E > \left\{ {\frac{hv}{1} + \frac{< E >}{D}} \right\}} \right\}.}}}}}}}} & (27)\end{matrix}$

The resulting expression, as usually written, where <ΔE²> is the meansquared Energy, <E> is the mean (average) Energy, <E_(v)>² is the squareof the Energy in the frequency range between v and υ+Δv, and V is thevolume of the blackbody radiator:

$\begin{matrix}{{< {\Delta\; E^{2}}>={\overset{\_}{E^{2}} - {\overset{\_}{E}}^{2}}} = {{hv} < E > {+ \frac{< E_{v} >^{2}}{D}}}} & (28) \\{< {\Delta\; E^{2}}>={hv} < E > {+ \frac{< E_{v} >^{2}}{8\pi\; V\frac{v^{2}}{c^{3}}\Delta\; v}}} & (29)\end{matrix}$

which contains two terms for the change in mean squared energy, one ofwhich Dr. Einstein identified with wave behaviour and one with particlebehaviour, leading to the uniquely his hypothesis of the wave-particleduality of light. The various arithmetic means and theoreticalincongruities deriving the fluctuation equation is provided in“Einstein's Fluctuation Formula. A Historical Overview” as well as“Reappraising Einstein's 1909 application of fluctuation theory toPlanckian radiation”. (Varró, 2006) (Irons, 2004) We simply note thatthe thought experiment conceived by Dr Einstein is of a perfectlyreflecting system inside a perfectly absorbing reservoir: a situationthat should prevent the specific system-radiation equilibrium envisionedfrom being established for a very long time. Nevertheless, if the systemis not perfectly reflecting, then it may take a long time, but thefluctuation states are still accessible as the system and the reservoirnear equilibrium. It should be clear that if these fluctuation statesonce reached from an equilibrium and then return back to the sameequilibrium, then the same states, if reached prior to the initialequilibrium, will lead to the same equilibrium. Perhaps the thoughtexperiment of Dr. Einstein, a blackbody radiator outside of a perfectlyreflecting radiator conversely parallels the blackbody radiator inside awhite body hohlraum of Prof. Wien, yet with the same conclusion:generally thermal radiation is of the same entropy regardless of thewall construction otherwise work could be extracted from two differentbodies at the same temperature—clearly an absurdity as expressed in thecharacteristic words (view) of Agassi.

By way of background, we note the statistical mathematics definition ofvariance of X is equal to the mean of the square of X minus the squareof the mean of X. The Einstein fluctuation formula—not withoutcontroversy—gives the variance (mean square deviation) of the energy ofblack-body radiation in a narrow spectral range in a sub-volume of acavity surrounded by perfectly reflecting walls. The formula is alsosaid to contain two terms as the “particle-term” (the Wien term) and the“wave-term” (the Rayleigh-Jeans term). (Irons, 2004)

The fluctuations are assumed to be a result of a divergence from andreturn to equilibrium, but perfectly reflecting walls should never reachthe equilibrium hypothesized as being operable. It has been postulatedrecently that blackbody radiation within a real cavity exhibits thethermal fluctuations predicted, but the fluctuations, with their “wave”and “particle” components, as argued, have their origin in the wallmaterial (the matter) and are not intrinsic to radiation. (Irons, 2004)This concept of duality being a characteristic of matter or theinteraction of matter with radiation, as opposed to an intrinsiccharacteristic of radiation itself is a long-based deliberation but isnot essential to resolve for the purpose of the instant invention. It isreported that a better received derivation of the Planckian model wasultimately provided in 1917. (Bacciagaluppi, 2017) (Einstein, On theQuantum Theory of Radiation, 1917)

Nevertheless, a more rigorous treatment for fluctuations of theradiation field remains unresolved to this date and the true meaning ofradiation duality is unclear: “there are two models of quantum phenomenaand both lead to equivalent results. One can start either from theclassical concept of a particle, or from classical wave concept andarrive at essentially the same quantum theory by quantizing one or theother”. (Kojevnikov, 2002) It is unclear in that such duality is said tobe a consequence solely of the measurement technique used and is not anintrinsic property—both simultaneously wave and particle—of radiation.

Still, the debate brings home the point, which is the instantinvention's main interest, that unless a cavity radiator is perfectlyreflecting, the rules governing cavity derived thermal radiation seem toapproximately apply even if not perfectly absorbing. It should also berepeated that a state which is accessible via fluctuations from anequilibrium state is also accessible for a system prior to reaching itsequilibrium state.

Material Indifference with Thermal Radiation. Why thermal radiation isindependent of the chemistry associated with the composition of thecavity radiator is of course not completely clear, especiallyconsidering the wave-particle duality may not emanate from thermalradiation itself but may have their origin in the wall material of thevacuum cavity radiator. (Irons, 2004) [The material used in the hohlraumradiators were initially platinum cylindrical enclosures lined with ironoxide and subsequently with chromium, nickel and cobalt oxides, andenclosed in a larger asbestos cylinder.] (Mehra & Rechenberg, 2001) Inthe instant invention, we use envelopes, glass cavities, mirrored ornot, which contain the phosphor upon which the primary radiation isincident. Whereas in a perfect black body design, the walls areperfectly black, the black body theory does not require that, as long asthe walls are not perfectly reflecting. Hence, we simulate a hohlraumusing coatings of phosphors.

There have been some arguments (hypotheses) to explain the lack of amolecular signature in thermal radiation. “The reason for cavityradiation having a characteristic spectrum which does not show thespecific fingerprint of the boundary material is the entropy of theclosed radiation system which takes a maximum value at the equilibriumconditions.” (Kabelac, 2012) This mechanism to transfer heat, whichmaximizes the entropy of the transfer as well as offering thecharacteristic energy of the transfer, negates or perhaps obscures thesignature of molecular and atomic interactions of the material thatcomprises the body of the cavity. In this case, the maximum entropyseems to reflect the maximum amount of information that is missing thatotherwise would allow one to describe the material that is inequilibrium with the radiation field (the maximum number of states thatare in equilibrium with the radiation field, information about which ismissing; hence all possible states are equally probable). (Giffin, 2008)(Jaynes, Information Theory and Statistical Mechanics., 1957)

Materials that are not perfect blackbodies may also reveal signatures ofelectronic transitions under thermal heating: “under a voltage bias,metals and semiconductors show thermal emissions emulating a blackbodywith additional emissions displaying a signature of “electron: hole”recombination unique to that metal or semiconductor“. (Liu, 2011)Thermal emission, clearly a general process for all matter, is perhapsonly “unique” in a perfect blackbody in that the appearance of molecularsignatures are completely obscured whereas thermal emission in generalmay include molecular signatures if the equilibrating material onlyapproximates a blackbody (i.e.; most materials are not blackbodies). Itis not clear if it is the equilibrium once attained that obscures thematerial's signature or if it is the benefit of obtaining maximumentropy—itself available only at equilibrium, or fluctuations therefrom.“Thermodynamically, a system in equilibrium maximizes entropy because wehave no information about its initial condition (except for conservedquantities): equipartition means no memory.” That a molecular signatureis normally represented by spectroscopic examination (i.e.;characterized by the electromagnetic radiation absorbed or emitted),itself not a conserved quantity, may be the reason that thermalradiation at equilibrium cannot reveal the characteristic of a perfectblackbody. [As noted by Mr. Barton, “we start by stressing the familiartruth, that no spectroscopic information specific to the system can beobtained as long as the system remains in overall thermal equilibrium:all that one then sees is the universal black-body spectrum.” ] (Barton,1987)

Thermodynamic Entropy. At its simplest, it is acknowledged that entropy,S, times temperature, T, has the units of energy. In traditionalthermodynamics, heat is in units of energy; work is in units of energy;“T×S” is in units of energy. In the thermodynamic equations of state,entropy is an important consideration and we are reminded of the two keythermodynamic equations for the functions called free energy, which isthe remaining energy free to perform work: the Gibbs free energy, G, andHelmholtz free energy, A (sometimes expressed as F), are in units ofenergy. The quintessential thermodynamic equations, in exactdifferential form, are:

dG=dH−TdS  (30):

and

dA=dE−TdS.  (31):

For a process in which energy is transferred both as work and heat, thelaw of conservation of energy says that the energy of the systems obeysthe equation

dE=δq+δw  (32):

where dE is the exact differential of internal energy; δq is the inexactdifferential of heat and δw is the inexact differential of work (pathdependent). All state functions are exact differentials (pathindependent). Thermal radiation transports heat; luminescence isradiation with a chemical potential. The present invention performsDown-Conversion at a temperature at which the chemical potential ofluminescence approaches zero, as we shall hereinafter demonstrate how toaccomplish.

In the integrated form, the exact differential, for a process in whichenergy is transferred both as work and heat, is

E=q+w  (33):

It can be observed that radiation exerts pressure on matter with whichit interacts, as radiation carries with it momentum, and the change inmomentum with time,

Δ(p)/Δ(t)=Δ(mv)/Δ(t)=mΔ(V)/Δ(t)=ma=F,  (34):

occurs from the force applied to the radiation by the surface and theopposing force applied by the radiation to the surface. However, undercustomary and usual (normal) terrestrial circumstances, radiationpressure is too small to be observed. Hence, our thermodynamic treatmentconsiders radiation of all types, thermal or luminescence, to beprincipally the transfer of energy (i.e.; the conservation of energy)and we perhaps safely ignore the requirements for transfer of momentum(i.e.; the conservation of momentum) (Bartoli, 1884) (Bradshaw, 2017)

This is not to say that the consideration of radiation pressure did notplay a role in the evolution of the interaction of radiation withmatter. As recent historical review has noted:

“By arranging black and reflecting cavities it was possible to constructa cyclic process in which heat was transmitted from one body to another.The cavity radiation could be handled by the usual methods ofthermodynamics. To remain consistent with the second law, one had toassume that radiation exerted a pressure so that a performance of workwould compensate for the transfer of heat.” (Badino, 2015)

Of course, the conservation of momentum is a fundamental law of physicswhich states that the momentum of a system is constant if there are noexternal forces acting on the system. It is embodied in Newton's firstlaw (the law of inertia). The law of conservation of energy states thatthe total energy of an isolated system remains constant, it is said tobe conserved over time. This law means that energy can neither becreated nor destroyed; rather, it can only be transformed or transferredfrom one form to another.

While it was long ago recognized that the field of thermal radiation wasa special case where thermodynamics of mechanical systems may be appliedto electromagnetic waves (or “photons”, whatever the case or one'spreference may be), it is generally accepted that a key step in thethermodynamic study of blackbody radiators was that embarked near 1900.

The thermodynamic treatment considers (for a cavity radiator that, whenheated, generated thermal radiation) an equilibrium interaction, betweenthe Radiated flux or “Radiated power” (energy per unit time) of amechanical resonator with an electromagnetic field, is generated. Thisequilibrium is expressed in the terms of power and spectral power as theaccompanying experiments would measure power and spectrum of thegenerated thermal radiation. This is interesting in that thermodynamicsdoes not have a time element associated with it (other than perhaps theSecond Law may be viewed as meaning “later”, which equates with “morerandom”). (Denbigh, 1953) Indeed, a core element of classicalthermodynamics is based on evaluating systems at equilibrium even thoughit is recognized that the thermodynamic system being analysed may takean exceedingly long time to reach equilibrium. (Brown, 2001)

More specifically, the equilibrium situation is the loss of oscillatorenergy to energy absorption by the electromagnetic field and back again:a good absorber (the oscillator) is a good emitter (the oscillator).Absorption of energy and emission of energy, back again, by theoscillating radiation field and the emission of energy and theabsorption of energy, back again, by the oscillators comprising thesurface is the essential equilibrium situation. Negating the importanceof the time element of Power, the equilibrium relationship then to bediscerned is between the energy of the oscillator (called resonators)and the energy of the thermal radiation.

In other words, the cavity radiator spectral distribution can beconsidered from the perspective of the electromagnetic field, or fromthe perspective of the oscillating electrons in the matter surroundingthe cavity. Because there is thermodynamic equilibrium between theradiation and the surrounding matter in a closed structure, the twoapproaches are equivalent. (Reiser & Schächter, 2013) If one considersthe oscillating electrons to form, in transient, partial radical ionnegative and partial radical ion positive charges at distances wherebythey lose correlation, then a magnetic field effect on blackbodyradiation is anticipated and indeed a small effect has been observed. Itis expected that the magnetic field effect would increase withincreasing temperature of the blackbody radiator.

At the time of the 1900's, the atomic view of matter was still unclearand hence the nature of the oscillators was not explored in a treatmentproffered by Dr. Planck; in his analyses: “ . . . he could not even sayfor certain what the oscillators were or why they were oscillating”.(Galison, 1981) [Indeed, many things about molecules that is presentlycommon knowledge was not known at the time of Dr. Planck's pioneeringwork in physics; see, for example, “It is still very uncertain”, Dr.Planck wrote in his Treatise on Thermodynamics, “whether the moleculesof liquid water are the same as those of ice. In fact, the anomalousproperties of water in the neighbourhood of its freezing point make itprobable that even in the liquid state its molecules are of differentkinds.”] (Planck, Treatise on Thermodynamics, 1969)

For the purpose of the instant invention, the Planckian model forblackbody radiators is critical in our treatment of Down-Conversion inthat it introduces the importance of the entropy of radiation, whereasall of the existing contemporaneous commentaries on Down-Conversion andsecondary radiation for the purpose of light emitting diode GeneralIllumination have focused on energy of radiation. The underlyingprinciple to exploit is the equilibrium of energy of the radiators (oroscillators) with the energy of the thermal radiation produced byheating the radiators (or oscillators) in the perfect blackbodyhohlraum. However, the same equilibrium that results in equating energymust also keep the entropy the same or increase with deviations from theequilibrium state: this is the Detailed Balance (Zhu & Fan, 2014) (for aprocess at equilibrium, the forward rate of each step is equal to thereverse rate of that step) which the instant invention shall hereinafterexploit.

More specifically, for the hohlraum, the equality is considered as theenergy of the oscillators in a certain state and the energy density ofthe radiation emanating from the volume of the blackbody radiator. Asmore elegantly explained, “At the end, there is a simple relationbetween the thermal radiation energy density u=u(v, T) and theelementary oscillator energy U (frequently denoted as U in the physicsliterature, although we often use the notation E). For example,“u_((radiation energy density))=[8πv²/c³]×U_((elementary oscillator energy)),a fundamental result in Planck's research. The simplest way tounderstand this relationship is by dimensional analysis, as U is anenergy, but u is energy density (energy per unit volume):

$\left\{ {\left( \frac{v}{c} \right)^{2}{d\left( \frac{v}{c} \right)}} \right\}$

is a differential inverse volume; 2×4π comes from polarization andangular integration.” (Boya, 2003)

While there are many forms of Planck's formula, we shall focus on energydensity where the general form for the energy density is

u(v,V)=Z _(V) U _(V)  (35):

where V is the volume of the hohlraum and v is the frequency of thespectral radiation. (Kramm & Mölders, Planck's blackbody radiation law:Presentation in different domains and determination of the relateddimensional constants., 2009) (Kramm & Herbert, Heuristic Derivation ofBlackbody Radiation Laws using Principles of Dimensional Analysis.,2008) The term Z_(V) is the spectral mode density of the radiation fieldenclosed in the hohlraum, in a cavity bounded by, in the words of Krammand Herbet, perfectly “reflecting” walls. [Again, we see the discrepancybetween perfectly reflecting walls and perfectly absorbing and thenemitting walls: this dichotomy is remindful of the model used by Feynmanto describe light scattering. In that model of Feynman, the scatteredlight is emitted light from decay of an electron from an excited stateto a ground state even though no chromophore for absorption is present.]

The spectral mode density has been found, by Dr. Planck and others,before his analyses and after, using classical physics and wavedescription of electromagnetic radiation:

$\begin{matrix}{Z_{V} = {\frac{8\pi\; v^{2}}{c^{3}}.}} & (36)\end{matrix}$

The spectral mode density is coupled with the average energy of theradiation to provide the energy density of radiation enclosed within thehohlraum:

$\begin{matrix}{{\mu\left( {v,V} \right)} = {{Z_{V}U_{V}} = {\frac{8\pi\; v^{2}}{c^{3}}\left\lbrack {{{hv}\text{/}e^{\frac{hv}{kT}}} - 1} \right\rbrack}}} & (37)\end{matrix}$

That above is the modern-day representation of the energy density withinthe cavity of the hohlraum using modern day constants, as opposed togeneric constants of integration. The symbol u is equivalent to the SIsymbol w_(e) for Radiant Energy Density and the symbol U is equivalentto

$\begin{matrix}{\left\lbrack \frac{\epsilon}{e^{\frac{\epsilon}{kT}} - 1} \right\rbrack = \left\lbrack \frac{hv}{e^{\frac{hv}{kT}} - 1} \right\rbrack} & (38)\end{matrix}$

with the Radiant Energy Density having the units of Joules per cubicmeter. (Varró, 2006)

A mathematical identity of interest is: (Hall, 2013)

$\begin{matrix}{\frac{1}{e^{x} - 1} = {{\sum\limits_{j = 1}^{\infty}\; e^{- {jx}}} = {e^{- x} + e^{{- 2}x} + e^{{- 3}x} + \cdots}}} & (39)\end{matrix}$

If we set

${x = \left\{ \frac{hv}{kT} \right\}},$

then (using the form to be integrated over all frequencies and V is thevolume of the cavity)

$\begin{matrix}{\mspace{76mu}{{E = {{\frac{8\pi\; v^{2}}{c^{3}}\left\lbrack {{{hv}\text{/}e^{\frac{hv}{kT}}} - 1} \right\rbrack}{Vdv}}}\mspace{76mu}{and}}} & (40) \\{E = {{{\frac{8\pi\; v^{2}}{c^{3}}\left\lbrack {{1\text{/}e^{\frac{hv}{kT}}} - 1} \right\rbrack}{hvVdv}} = {\frac{8\pi\;{hv}^{3}{Vdv}}{c^{3}}\left\lbrack {e^{- \frac{hv}{kT}} + e^{{- 2}{hv}\text{/}{kT}} + \cdots} \right\rbrack}}} & (41)\end{matrix}$

Thermodynamics of Thermal Radiation—The Entropy of Radiation. Thethermodynamic evaluations of mechanical systems were well known at thetime of Dr. Planck and the first and second derivatives of variousthermodynamic relationships was also clear and of known importance. “Animportant quantity in thermodynamics is the second derivative of theentropy, this is closely related to the Second Law: negativity of thesecond derivative guarantees that the entropy will increase back toequilibrium if the system is disturbed.” (Boya, 2003)

Perhaps the first introduction of radiation entropy is, it has beendemonstrated, by a contemporary of Dr. Planck:

“(However,) Wien noticed that if a cavity full of radiation is a genuinethermal system, it must be possible to assign a temperature, and even anentropy, to it . . . ”. For the black-body radiation, all components (atdifferent wavelength, polarization state, and direction of propagation)have the same temperature, although the energy density differs. Theuniformity of the temperature, Wien noticed, derives from the fact that“these components are independent from one another, because we canproduce a radiation that contains only one component.” This suggeststhat a cavity radiation is analogous to a mixture of gases of dissimilarchemical species.”

Why this is of importance—the second derivative to reflect a maximum—isnot so clear other than to support the hypothesis that thermal radiationis an electromagnetic field of a type that has the maximum entropycarried by it, versus non-thermal electromagnetic fields that carry lessentropy than the maximum. The significance of this would be, perhaps,associated with the usual fluctuations of a system normally atequilibrium (as in the fluctuation theory developed by Dr. Einstein):negativity of the second derivative guarantees that the entropy willincrease back to equilibrium if the system is disturbed. The historicalimpetus for considering that fluctuations of cavity radiation, normallyat equilibrium, should necessitate a return to maximum entropy for theradiation emanating therefrom is driven by the fact that thermalradiation is defined at an equilibrium state: thermodynamic reactions atequilibrium have the maximum entropy (and Gibbs free energy at zero andHelmholtz free energy at a minimum). This was a fundamental assumptionintroduced by Dr. Planck and equilibrium thermodynamics provided a pathforward using simply second derivatives of entropy to energy.

It is helpful to recall some simple consequences of derivatives andsecond derivatives:

If

$\begin{matrix}{{{\frac{df}{dx}(p)} = 0}{and}} & (42) \\{{\frac{d^{2}f}{{dx}^{2}}(p)} > 0} & (43)\end{matrix}$

then f(x) has a local minimum at x=p.

If

$\begin{matrix}{{{\frac{df}{dx}(p)} = 0}{and}} & (44) \\{{\frac{d^{2}f}{{dx}^{2}}(p)} < 0} & (45)\end{matrix}$

then f(x) has a local maximum at x=p.

If

$\begin{matrix}{{{\frac{df}{dx}(p)} = 0}{and}} & (46) \\{{\frac{d^{2}f}{{dx}^{2}}(p)} = 0} & (47)\end{matrix}$

then we learn no new information about the behaviour of f(x) at x=p.

Second derivatives and mixed partial derivatives have a long history ofmathematical evaluations and proofs to demonstrate equivalency. Secondderivatives are instrumental in wave mechanics, the classical waveequation, a partial differential equation where x and t are independentvariables and μ a dependent variable, is (with υ being velocity andvλ=υ; and ω=2πv):

$\begin{matrix}{\frac{d^{2}u}{{dx}^{2}} = {\frac{1}{v^{2}}\frac{d^{2}u}{{dt}^{2}}\overset{yields}{\rightarrow}}} & (48) \\{{u\left( {x,t} \right)} = {{{\psi(x)}{f(t)}}\overset{yields}{\rightarrow}}} & (49) \\{\frac{d^{2}{\psi(x)}}{{dx}^{2}} = {\left( {- \frac{\omega^{2}}{v^{2}}} \right){\psi(x)}}} & (50)\end{matrix}$

where the standard wave equation used is f (t)=e^(iωt) [Euler's formulastates that e^(ix)=cos x+i sin x] and which ultimately and simply leadsto Schrödinger's time-independent wavefunction with appropriatesubstitutions for de Broglie's momentum. (McQuarrie, 2007)

In thermodynamics, because of Maxwell Relations, there is almost aninfinite set of second derivatives or partial derivatives, some withsignificant physical meaning. With respect to the state functionentropy, the Principle of Maximum Entropy states that at equilibrium,dS=0 and d²S<0. We shall solely use second derivatives of entropy asother justifications originally advanced are supportive of, but notnecessary to be and generally were not included within, the mathematicaltreatment that yields the energy density of the radiation enclosedwithin the hohlraum. The Radiance associated therewith and the spectraldistribution thereof, of course, are the characteristics of theradiation that exits the hohlraum, to be used in the universe toperformance work and to transfer heat (some of which may be convertedback to perform work using a heat engine).

If one considers a thermodynamic system's energy, U, based on theentropy, S, the volume, V and the number of “particles, N, U_(S,V,N),there are nine second order differentials, only six of which are unique:one that is unique is

$\left\lbrack \frac{d^{2}U}{{ds}^{2}} \right\rbrack.$

The mathematics associated with the entropy to energy treatment ofblackbody cavity radiators is essentially an integration of adifferential equation. A starting point for this approach may beconsidered in a very simple form whereby the negative sign indicates anentropic maximum, as is required in the theory of Dr. Planck:

$\begin{matrix}{{- \left\lbrack \frac{d^{2}S}{{dU}^{2}} \right\rbrack^{- 1}} = {C \cdot v \cdot {U.}}} & (51)\end{matrix}$

where C is a constant not otherwise defined and U and S are the energyand the entropy of the resonator (oscillator) of the blackbody thatinteracts with (and in equilibrium with) the electromagnetic fieldenclosed therein and v is the radiation frequency. (Martinelli, 2017) Atsome point in the arithmetic model, energy U and entropy S of theradiators must become the energy and the entropy of the thermalradiation. Consequently, this second differential equation, at theleast, assured that the entropy (of the thermal radiation) is a maximum,for a particular relevant energy of the oscillator, in equilibrium withthe thermal radiation. As we now know, the energy of radiation is afunction of a Constant times the frequency. Hence, the seconddifferential used as a starting point will be a function of the squareof the energy as U is also included at the starting point. Since thesecond differential is a function only of S and U, at this point thefrequency is a variable not changed. However, to obtain the total energyand total entropy, one would then have to integrate to dv afterobtaining the energy and entropy at only one v.

Parenthetically, it is noted that the first derivative

$\begin{matrix}{\frac{dS}{dU} = {\frac{1}{T} = \frac{dS}{dE}}} & (52)\end{matrix}$

is well known as the thermodynamic temperature, T (and the symbol “U” isredundant with the symbol “E”). (Boya, 2003) The utility of thethermodynamic temperature is that it provides an opportunity to changethe evaluation of the experimental results from discerning arelationship between frequency and temperature (those which are measuredexperimentally) to distinguishing a relationship between entropy andenergy. For a cavity radiator, the temperature at which thermalradiation is in equilibrium with the matter (material) that is radiatingis clear. It is not always clear what is the entropy and energy of thesame system. It is noted that “(T) temperature is fundamentally astatistical concept requiring many bodies. If the distribution ofenergies is broad, the temperature is high. If the distribution ofenergies is narrow, the temperature is low. Temperature is basically ameasure of the half-width of the distribution”. (Rabinowitz, 2006)

The thermodynamic temperature equation allows for the model of thehohlraum at a particular temperature (which emits a characteristicspectrum of frequencies of radiation) in an experimental volume of thecavity itself, is what links entropy and energy and more importantly,entropy density and energy density. Thus the search for the solutions tothe second differential, which ensures entropy is a maximum to theenergy, will yield a relationship of entropy in terms of frequency andenergy and energy in terms of entropy and frequency, expressed in theforms of density (to remove the Volume sensitivity to the relationship).When integrated over all frequencies, the solution to the seconddifferential then becomes extended to total energy density and totalentropy density. The fundamental question then becomes what is themathematical expression of entropy to energy that is accurately definingthe second differential of their relationship.

Other well-known thermodynamic relationships are:

$\begin{matrix}{{- \left\lbrack \frac{d^{2}S}{{dU}^{2}} \right\rbrack^{- 1}} = {\frac{d^{2}A}{{dT}^{2}} = {{- \frac{dS}{dT}} = {\frac{C_{V}}{T} = {\frac{1}{T}\frac{dU}{dT}}}}}} & (53)\end{matrix}$

Note that

${C_{V} = {T\left( \frac{\partial S}{\partial T} \right)}_{V,N}};$

in addition

${C_{V} = {\left( \frac{\partial U}{\partial T} \right)_{V,N} = \left( \frac{\partial E}{\partial T} \right)_{V,N}}},$

as the symbol “U” is redundant with the symbol “E”.

The thermodynamic temperature being expressed in terms of entropy andenergy is an identity that appears in many theoretical treatments todevelop models for energy and, perhaps, is a major motivation forseeking, by experiment or by model, an equilibrium situation among thesystem parameters.

Please note that the heat capacity at constant volume is C_(V) whereasthe term C·v is a to be defined constant C multiplied by the frequencyv. Note that an essentially equivalent relationship was written in“Planck, Photon Statistics, and Bose-Einstein Condensation” as shown:

$\begin{matrix}{{--\left\lbrack \frac{d^{2}S}{{dU}^{2}} \right\rbrack^{- 1}} = {{{C \cdot v \cdot U}\overset{yields}{\rightarrow}\left\lbrack \frac{d^{2}S}{{dE}^{2}} \right\rbrack} = {- \frac{1}{{bv}\;\epsilon}}}} & (54)\end{matrix}$

Also, see Equation 2.13 in Statistical Physics: A Probabilistic Approachby Bernhard Lavenda. (Lavenda, 1991)

The nomenclature symbol “E” is frequently used instead of “U”; and where“ϵ” is energy of the oscillator.

Starting from

$\begin{matrix}{{- \left\lbrack \frac{d^{2}S}{{dU}^{2}} \right\rbrack^{- 1}} = {C \cdot v \cdot U}} & (55)\end{matrix}$

we consider with some minor algebraic reformulation,

$\begin{matrix}{{\int{\left( \frac{dS}{dU} \right)\left( \frac{dS}{dU} \right){dU}}} = {{- \left( \frac{1}{C \cdot v} \right)}{\int{\left( \frac{1}{U} \right){dU}}}}} & (56)\end{matrix}$

which is then integrated to yield

$\begin{matrix}{\left( \frac{dS}{dU} \right) = {{- \left( \frac{1}{C \cdot v} \right)}\mspace{14mu}\left( {{\log_{e}\mspace{14mu} U} + b_{1}} \right)}} & (57)\end{matrix}$

The term b₁ is the arbitrary constant of integration not otherwisedefined.

Consequently, rearranging

$\begin{matrix}{{dS} = {{- \frac{1}{C \cdot v}}\mspace{14mu}\left( {{\log_{e}\mspace{14mu} U} + b_{1}} \right)\mspace{14mu}{dU}}} & (58)\end{matrix}$

and inserting the definition for the thermodynamic temperature,

dS=T ⁻¹ dU  (59):

which follows from the well-known thermodynamic relation

$\begin{matrix}{\frac{dS}{dU} = \frac{1}{T}} & (60)\end{matrix}$

Setting dS equalities consequently yields

$\begin{matrix}{{{- \left( \frac{1}{C \cdot v} \right)}\mspace{14mu}\left( {{\log_{e}\mspace{14mu} U} + b_{1}} \right){dU}} = {T^{- 1}\mspace{14mu}{dU}}} & (61)\end{matrix}$

and, consecutively,

(log_(e) U+b ₁)(dU)=−C·vT ⁻¹ dU  (62):

(log_(e) U+b ₁)=−C·vT ⁻¹  (63):

(log_(e) U)=−[C·vT ⁻¹ +b ₁]  (64):

e ^(log) ^(e) ^(U) =e ^(−[C·vT) ⁻¹ ^(−b) ¹ ^(])=(e ^(b) ¹ )(e ^(−C·vT)⁻¹ ).  (65):

One can then solve arithmetically for U, by rearranging and settingthese terms to be exponents of e. Thus,

$\begin{matrix}{e^{\log_{e}\mspace{14mu} U} = {U = {{K\left\lbrack \frac{1}{e^{\frac{Cv}{T}}} \right\rbrack} = {Ke}^{- \frac{C \cdot v}{T}}}}} & (66)\end{matrix}$

where K is an integration constant, e^(b) ¹ =K, and C·v is the functionof frequency (constant C times the frequency, v) from the initial seconddifferential equation relating entropy to energy. The form of thisequation is Wien's approximation (also sometimes called Wien's law orthe Wien distribution law), determined by Mr. Wien solely on anempirical basis. In modern days terminology, Wien's distribution law iswritten as (where

$\left. {{C \cdot v} = {\frac{h}{k}v}} \right).$

$\begin{matrix}{{L\left( {v,T} \right)} = {\left\lbrack \frac{2{hv}^{3}}{c^{2}} \right\rbrack e^{- \frac{hv}{kT}}}} & (67)\end{matrix}$

Note that at this point there is no need to introduce any quantum ofdiscrete energy for the oscillators (nor for the radiation itself thatresults from energy transfer by and between the oscillators and theradiation field). Clearly the constant C in the mathematical approach ispresently equated with the constant h/k, or Planck's constant divided byBoltzmann's constant. The original second-order differential equationcould then have been written as:

$\begin{matrix}{{- \left\lbrack \frac{d^{2}S}{{dU}^{2}} \right\rbrack^{- 1}} = \frac{hvU}{k}} & (68)\end{matrix}$

where the constant C is

$C = {\frac{h}{k}.}$

[In Kuhn, β is the symbol used to represent

$\frac{h}{k},$

which is the present constant C.] (Kuhn T. S., 1978) The omnipresentexponential term then originates from the integration of the 1/U term;itself of such a form due to the relationship of both

$\begin{matrix}{{\frac{dS}{dU} = \frac{1}{T}};{{{and}\mspace{14mu}\frac{d^{2}S}{{dU}^{2}}} \propto {\frac{1}{U}.}}} & (69)\end{matrix}$

The simple second derivative form presented herein can be found in theclassic works of others (that focus on the historical developments ofthe quantum by Planck) which we present herein for completeness:

$\begin{matrix}{{- \left\lbrack \frac{d^{2}S}{{dU}^{2}} \right\rbrack^{- 1}} = {{C \cdot v \cdot U} = \frac{hvU}{k}}} & (70)\end{matrix}$

where C is multiplied by v and by U

$\begin{matrix}{{\frac{d^{2}S}{{dU}^{2}} = {- \frac{\alpha}{U}}};\alpha} & (71)\end{matrix}$

not defined (Kuhn T. S., 1978) page 96

${\left\lbrack \frac{d^{2}S}{{dU}^{2}} \right\rbrack^{1} = {{- \frac{1}{\gamma\; v\overset{\_}{E}}}\mspace{14mu}\left( {{Lavenda},1991} \right)}},{{page}\mspace{14mu} 75}$

Dr. Planck had offered a look at how the “energy” of electromagneticradiation (units not yet known; hence in what form are the units ofenergy is not yet determined) is related to the energy of oscillators inthermal equilibrium with the electromagnetic radiation such that the“energy” of both are simply a function of the temperature T of thecavity radiator. It appears that by considering the second derivative ofentropy of the oscillators to the energy of the oscillators, ensuringthe total entropy is a maximum, Dr. Planck showed how one can thenmathematically convert same to determine the “energy” of the thermalradiation.

At this point, a better starting point might be with two constants, Cand D, is considered which we have expressed as a power series:

$\begin{matrix}{{- \left( \frac{d^{2}S}{{dU}^{2}} \right)^{- 1}} = {{C \cdot v \cdot U} + {DU}^{2}}} & (73)\end{matrix}$

We again note that the negative sign is required in order for theentropy to be a maximum.

[In Joseph Agassi's Appen, the second differential equation does nothave a negative sign and is of the form

$\left( \frac{d^{2}S}{{dU}^{2}} \right) = {\frac{a}{U\left( {b + U} \right)}\mspace{14mu}\left( {{see}\mspace{14mu}{page}\mspace{14mu} 128} \right)\mspace{14mu}\left( {{Agassi},1993} \right)}$

Without the negative sign, the double integration is reported as(without explanation) as

$U_{v} = {{{hv}\left( {e^{\frac{hv}{kT}} - 1} \right)}^{- 1}\mspace{14mu}\left( {{see}\mspace{14mu}{Equations}\mspace{14mu}(14)\mspace{14mu}{and}\mspace{14mu}(15)} \right)}$

which is not an identity with Planck's formula.]

Historical teachings focus on the following set of second differentialequations:

$\begin{matrix}{{\left( \frac{d^{2}S}{{dU}^{2}} \right) = {{- \frac{\alpha}{U\left( {\beta + U} \right)}}\mspace{14mu}\left( {{{Kuhn}\mspace{14mu}{T.S.}},1978} \right)}},{{page}\mspace{14mu} 97}} & (76)\end{matrix}$

and Planck stated that it yielded

$\begin{matrix}{E = \frac{C\;\lambda^{- 5}}{e^{\frac{C}{\lambda\; T}} - 1}} & (77)\end{matrix}$

where one gets a radiation formula with two constants, c and C, “which,as far as I can see at the moment, fits the observational data,published up to now, as satisfactory as the best equations put forwardfor the spectrum, namely those of Thiesen, Lummer-Jahnke, andLummer-Pringsheim”.]

Our starting equation can be re-written to allow for an evaluation thatuses a (similar) starting point articulated in other reviews (butmissing the actual mathematical proof) which we show for completenesspurposes:

$\begin{matrix}{\frac{d^{2}S}{{dU}^{2}} = {{- \frac{1}{{CvU} + {DU}^{2}}} = {- \frac{\alpha}{U\left( {{\beta\; v} + U} \right)}}}} & (78)\end{matrix}$

where we arbitrarily assign the constant C to be

$\begin{matrix}{C = \frac{\beta}{\alpha}} & (79)\end{matrix}$

and constant D to be

$\begin{matrix}{D = {\frac{1}{\alpha}\mspace{14mu}\left( {{i.e.};{\frac{C}{D} = \beta}} \right)}} & (80)\end{matrix}$

or as expressed by

$\begin{matrix}{\frac{d^{2}S}{{dU}^{2}} = {- \frac{\alpha}{U\left( {{\beta\; v} + U} \right)}}} & (81)\end{matrix}$

Note in Boya the starting expression: (Boya, 2003)

$\begin{matrix}{\left\{ \left( \left( \frac{d^{2}S}{{dU}^{2}} \right)^{- 1} \right) \right\} = {{\alpha\; U} + {\beta\; U^{2}}}} & (82)\end{matrix}$

and Parisi

$\begin{matrix}{\frac{d^{2}S}{{dU}^{2}} = {- \frac{\alpha(v)}{U\left( {{\beta(v)} + U} \right)}}} & (83)\end{matrix}$

Note that the U² now introduced (in comparison with initial startingpoint) implies, according to Parisi,

$\begin{matrix}{\frac{d^{2}S}{{dU}^{2}} \propto U^{- 2}} & (84)\end{matrix}$

as for a harmonic oscillator as opposed to the initially proposedrelationship of

$\begin{matrix}{\frac{d^{2}S}{{dU}^{2}} \propto U^{- 1}} & (85)\end{matrix}$

[See Mr. Martinelli's work where an alternative starting point isprovided:

$\begin{matrix}{\frac{d^{2}S}{{dU}^{2}} = {- \frac{a}{{avU}\left( {{bv} + U} \right)}}} & (86)\end{matrix}$

where the integration is said to yield the term

bv/(e^(−αv)/T)−] (Martinelli, 2017)

Note the mathematical manipulation of

$\begin{matrix}{U = {{< E>=\overset{\_}{E}} = {\frac{hvx}{1 - x} = {\frac{hv}{x^{- 1} - 1} = \frac{hv}{e^{\frac{hv}{kT}} - 1}}}}} & (87)\end{matrix}$

if one sets

$x = {e^{- \frac{hv}{kT}}.}$

The second differential provided by Martinelli in his starting point isnegative, as one would require for the entropy to be a maximum. However,he provides without mention the same end formula as Aggasi. Without thenegative sign, the double integration is reported as (withoutexplanation):

$\begin{matrix}{U_{v} = {{hv}\left( {e^{\frac{- {hv}}{kT}} - 1} \right)}^{- 1}} & (88)\end{matrix}$

(see Agassi's Equations (14) and (15))

which is not an identity with Planck's formula.

From our proposed starting point, we rearrange and then integrate:

$\begin{matrix}{\frac{d^{2}S}{{dU}^{2}} = {{- \frac{\alpha}{U\left( {{\beta\; v} + U} \right)}} = {{- \left( \frac{\alpha}{\beta\; v} \right)}\left\{ {\frac{1}{U} - \frac{1}{{\beta\; v} + U}} \right\}}}} & (89)\end{matrix}$

Note the algebraic rearrangement provides an equivalent representationby adding U and then subtracting U:

$\begin{matrix}{{- \frac{\alpha}{U\left( {{\beta\; v} + U} \right)}} = {{{- \left( \frac{\alpha}{\beta\; v} \right)}\left\{ \frac{\left( {\beta\; v} \right)}{U\left( {{\beta\; v} + U} \right)} \right\}} = {{{- \left( \frac{\alpha}{\beta\; v} \right)}\left\{ \frac{\left( {{\beta\; v} + U} \right) - U}{U\left( {{\beta\; v} + U} \right)} \right\}} = {{- \left( \frac{\alpha}{\beta\; v} \right)}\left\{ {\frac{1}{U} - \frac{1}{{\beta\; v} + U}} \right\}}}}} & (90)\end{matrix}$

Then, we integrate and set equal to the thermodynamic temperature:

$\begin{matrix}{{\int{\frac{dS}{dU}\frac{dS}{dU}{dU}}} = {{- \left( \frac{\alpha}{\beta\; v} \right)}{\int{\left\{ {\frac{1}{U} - \frac{1}{{\beta\; v} + U}} \right\}{dU}}}}} & (91)\end{matrix}$

Note that the mathematical integration of the form below is known to be:

$\begin{matrix}{{{\int{\left\{ {\frac{1}{x} - \frac{1}{c + x}} \right\}{dx}}} = {{{\int{\frac{1}{x}{dx}}} - {\int{\frac{1}{c + x}{dx}}}} = {{\ln(x)} - {\ln\left( {c + x} \right)} + C}}}\mspace{76mu}{Hence}} & (92) \\{\mspace{76mu}{{{- \left( \frac{\alpha}{\beta\; v} \right)}{\int{\left\{ {\frac{1}{U} - \frac{1}{{\beta\; v} + U}} \right\}{dU}}}} = {{- \left( \frac{\alpha}{\beta\; v} \right)}\left\{ {{\ln(U)} - {\ln\left( {{\beta\; v} + U} \right)}} \right\}}}} & (93) \\{{\frac{dS}{dU} = {\frac{1}{T} = {{{- \left( \frac{\alpha}{\beta\; v} \right)}\left\{ {{\ln(U)} - {\ln\left( {{\beta\; v} + U} \right)}} \right\}} = {{\left( {+ \frac{\alpha}{\beta\; v}} \right)\left\{ {{- {\ln(U)}} + {\ln\left( {{\beta\; v} + U} \right)}} \right\}} = {\frac{\alpha}{\beta\; v}\left\{ {\ln\frac{{\beta\; v} + U}{U}} \right\}}}}}}\mspace{76mu}{then}} & (94) \\{\mspace{76mu}{\frac{\beta\; v}{\alpha\; T} = \left\{ {\ln\frac{{\beta\; v} + U}{U}} \right\}}} & (95) \\{\mspace{76mu}{{e^{\frac{\beta\; v}{\alpha\; T} =}\left\{ \frac{U + {\beta\; v}}{U} \right\}} = {1 + \frac{bv}{U}}}} & (96) \\{\mspace{76mu}{{e^{\frac{\beta\; v}{\alpha\; T}} - 1} = \frac{bv}{U}}} & (97) \\{\mspace{76mu}{\frac{\beta\; v}{e^{\frac{\beta\; v}{\alpha\; T}} - 1} = {U = \frac{hv}{e^{\frac{hv}{kT}} - 1}}}} & (98) \\{\mspace{76mu}{\frac{\frac{C}{D}v}{e^{\frac{Cv}{T}} - 1} = U}} & (99)\end{matrix}$

The starting point proposed in the instant invention, therefore, is:

$\begin{matrix}{{- \left( \frac{d^{2}S}{{dU}^{2}} \right)^{- 1}} = {{\frac{hv}{k}U} + {\frac{1}{k}U^{2}}}} & (100)\end{matrix}$

with the units being Joules per K. The starting point proposed byMartinelli (2017) would be (with units being Joules² per K):

$\begin{matrix}{{- \left( \frac{d^{2}S}{{dU}^{2}} \right)^{- 1}} = {{{{abv}^{2}U} + {avU}^{2}} = {{hv}\left\{ {{\frac{({hv})^{1}}{k}U} + {\frac{1}{k}U^{2}}} \right\}}}} & (101)\end{matrix}$

The units for Planck's constant h (h=β) is joule-seconds; for frequencyv, reciprocal seconds; for Boltzmann's constant (k=α), joules perKelvin. The constant C is

${C = \frac{h}{k}};$

the constant D is k−1.

Then calculated the energy U is (this is the usual written form ofPlanck's formula):

$\begin{matrix}{U = \left\lbrack {{hv}\text{/}\left( {e^{\frac{hv}{kT}} - 1} \right)} \right\rbrack} & (102)\end{matrix}$

with the units for the “energy” U in joules.

The expression derived by Martinelli is: (Martinelli, 2017)

U={bv/[(e ^(−αv) /T)−1]=hv/[(e ^(−hv) /kT)−]}  (103):

One notes that there is nothing quantum mechanical about this treatmentor required in the mathematical derivation at this point other than toassign a constant the symbol, h, which later on would be found to be acritical constant in quantum mechanics.

Therefore, classical thermodynamic treatment of entropy as a function ofenergy, provides an equation that is in near perfect agreement with theexperimental measurements associated with energy, power and intensity.This experimental verification confirms that for a given energy(frequency), the entropy of thermal radiation is a maximum. One can theninfer that for a given frequency from radiation that is not thermalradiation, the entropy is less than a maximum and only approaches amaximum if the radiation becomes more thermal like and the processapproaches an equilibrium. Fluctuations away from equilibrium allow areturn to equilibrium; hence approaching an equilibrium from the otherside means at some time thereinafter an equilibrium will be reached.

As noted, before, a mathematical identity of interest is

$\begin{matrix}{\frac{1}{e^{x} - 1} = {{\sum\limits_{j = 1}^{\infty}\; e^{- {jx}}} = {e^{- x} + e^{{- 2}x} + e^{{- 3}x} + \cdots}}} & (103)\end{matrix}$

If we set

${x = \frac{hv}{kT}},$

then

$\begin{matrix}{\mspace{76mu}{E = {{\frac{8\pi\; v^{2}}{c^{3}}\left\lbrack {{{hv}\text{/}e^{\frac{hv}{kT}}} - 1} \right\rbrack}{Vdv}}}} & (105) \\{E = {{{\frac{8\pi\; v^{2}}{c^{3}}\left\lbrack {{1\text{/}e^{\frac{hv}{kT}}} - 1} \right\rbrack}{hvVdv}} = {\frac{8\pi\;{hv}^{3}{Vdv}}{c^{3}}\left\lbrack {e^{- \frac{hv}{kT}} + e^{{- 2}{hv}\text{/}{kT}} + \cdots} \right\rbrack}}} & (106)\end{matrix}$

so that the energy of each frequency state is ∈=nhv,n=1, 2, 3, 4 . . . .

If one sets includes the number of modes in the blackbody, one obtainsthe Planck Radiation Law in terms of frequency and where we use as aconstant Planck's constant and Boltzmann's constant (in modern daynotation):

$\begin{matrix}{{{Radiance}\mspace{14mu}{L_{v}\left( {v,T} \right)}} = {{I\left( {v,T} \right)} = {\left\lbrack \frac{2v^{2}}{c^{3}} \right\rbrack\left\lbrack \frac{hv}{e^{\frac{hv}{lT}} - 1} \right\rbrack}}} & (107)\end{matrix}$

This is called Radiance and has units of Joules per second per steradianper square meter.

Planck's Radiation Law can be expressed in different formulations thatare all equivalent. Further, Planck's law can be expressed in terms ofRadiance, Radiant Flux Density or Radiant Energy Density, among otherpossible expressions (with units shown):

$\begin{matrix}{{{Radiance}\mspace{14mu}{L_{v}\left( {v,T} \right)}} = {{I\left( {v,T} \right)} = {{\left\lbrack \frac{2{hv}^{3}}{c^{2}} \right\rbrack\left\lbrack \frac{1}{e^{\frac{hv}{lT}} - 1} \right\rbrack} = {{\left\lbrack \frac{2v^{2}}{c^{2}} \right\rbrack\left\lbrack \frac{hv}{e^{\frac{hv}{lT}} - 1} \right\rbrack}W\text{/}m^{2}{sr}}}}} & (108) \\{{{Irradiance}\mspace{14mu}{or}\mspace{14mu}{Radiant}\mspace{14mu}{Flux}\mspace{14mu}{Density}\mspace{14mu}{E_{v}\left( {v,T} \right)}} = {{\left\lbrack \frac{2v^{2}}{c^{2}} \right\rbrack\left\lbrack \frac{hv}{e^{\frac{hv}{lT}} - 1} \right\rbrack}W\text{/}m^{2}}} & (109) \\{{{Radiant}\mspace{14mu}{Energy}\mspace{14mu}{Density}\mspace{14mu}{w_{e}\left( {v,T} \right)}} = {{\left\lbrack \frac{8\pi\; v^{2}}{c^{3}} \right\rbrack\left\lbrack \frac{hv}{e^{\frac{hv}{lT}} - 1} \right\rbrack}{Joules}\text{/}m^{3}}} & (110)\end{matrix}$

The full entropy density, in terms of the “energy”, of the thermalradiation, are: (Boya, 2003)

$\begin{matrix}{{{S(U)} = {k\left\lbrack {{\ln\left( {1 + {U\text{/}{hv}}} \right)}^{({1 + {U\text{/}{hv}}})} - {\ln\left( \left( {U\text{/}{hv}} \right)^{({U\text{/}{hv}})} \right)}} \right\rbrack}}{or}} & (111) \\{{S(U)} = {k\left\lbrack {{\left( {1 + {U\text{/}{hv}}} \right){\ln\left( {1 + {U\text{/}{hv}}} \right)}^{(1)}} - {\left( {U\text{/}{hv}} \right){\ln\left( {U\text{/}{hv}} \right)}^{(1)}}} \right\rbrack}} & (112)\end{matrix}$

The term

$\frac{U}{hv}$

is:

$\begin{matrix}{\frac{hv}{\frac{\left( {e^{\frac{hv}{kT}} - 1} \right)}{hv}} = \frac{1}{\left( {e^{\frac{hv}{kT}} - 1} \right)}} & (113)\end{matrix}$

which is the number of photons at frequency v (i.e.; n_(p)) and which isunitless. It is also, interestingly, the average quantum number of aharmonic oscillator.

Therefore, the entropy density term is in the form where

S(U)=k[(1+n _(p))ln(1+n _(p))−(n _(p))ln(n _(p))]  (114):

and the change in entropy in terms of n_(p) is

$\begin{matrix}{\frac{dS}{{dn}_{p}} = {{k\left\lbrack {{\ln\left( {1 + n_{p}} \right)} - {\ln\left( n_{p} \right)}} \right\rbrack} = {k\left\lbrack {{\ln\left( \frac{1 + n_{p}}{n_{p}} \right)} = {k\left\lbrack {\ln\left( {1 + \frac{1}{n_{p}}} \right)} \right\rbrack}} \right.}}} & (115)\end{matrix}$

The entropy density could have been calculated from the energy densityfunction derived by Planck but appears not have been done so. Rather acombinatorial consideration of oscillators (called complexions byPlanck) led him to the entropy density function whose first derivativewith respect to energy was consistent with the energy density functioncontained. The complete derivation has been reported elsewhere.Historical reviews seem also to not provide an explicit derivationentropy density from the thermodynamic definition of temperature or eventhe initially hypothesized second derivatives. See “Thermodynamics ofradiation pressure and photon momentum” and “Radiation entropy flux andentropy production of the Earth system” for discussions on thecalculation of the entropy density for a blackbody radiator. (Mansuripur& Han, 10'7) (Wu & Liu, 2010) Note that Appendix A by Wu & Liu providesthe calculus to derive the entropy density equation from the energydensity equation, referencing “Gradshteyn, I. S., and I. M. Ryzhik(1980), Tables of Integrals, Series and Products, 1160 pp., Academic,New York”, and hence is referenced herein by attribution.

The entropy maximum in terms of frequency can be determined at anyparticular temperature of the hohlraum; it is generally at a shorterfrequency than the spectral maximum in terms of Radiance (energy). Thus,the region (in radiation frequencies) bathochromic to the entropicmaximum provides greater radiation entropy relative to radiation energythan that region hypsochromic to the energy maximum.

What is unique about Planck's Radiation Law is the fact that theright-hand side is independent of the geometry or the properties of thebody. As such, many consider it as a fundamental law and in this regardit as an upper limit to what a body can emit as thermal radiation, allother bodies, regardless of geometry and material properties would yieldless entropy and less energy. (Reiser & Schächter, 2013)

There are two important points regarding Planck's Radiation Law: when itis integrated over all frequencies (or wavelengths) to provide the totalenergy of radiation enclosed within the vacuum cavity radiator, ityields the Stefan-Boltzman law for T4 dependency. When it, Planck'sRadiation Law, is evaluated at those frequencies (or wavelengths) forwhich Wien's approximation (also sometimes called Wien's law or the Wiendistribution law) is found to be valid, Planck's distribution law infact reduces to that of Wien.

When integrated over all wavelengths emitted, the thermodynamicrelations of state for a photon gas are determined and are now wellsummarized in contemporaneous literature. The photon gas is a model of aquantum mechanical system of quanta of an electro-magnetic field withacceptance of the particle-wave duality of such a field, such dualityemanating from the fluctuation analysis of Dr. Einstein.

More specifically, when hv>>kT, then in terms of Radiance (Joules persecond per steradian per square meter)

$\begin{matrix}{{L\left( {v,T} \right)} = {\left\lbrack \frac{2{hv}^{3}}{c^{2}} \right\rbrack\left\lbrack \frac{1}{e^{\frac{hv}{lT}} - 1} \right\rbrack}} & (116)\end{matrix}$

becomes

$\begin{matrix}{{L\left( {v,T} \right)} = {\left\lbrack \frac{2{hv}^{3}}{c^{2}} \right\rbrack\left\lbrack \frac{1}{e^{\frac{hv}{lT}}} \right\rbrack}} & (117)\end{matrix}$

We note that Wien's approximation (also sometimes called Wien's law orthe Wien distribution law) is valid at high frequencies (high energy)and in that case hv>>kT. At low frequencies (low energy), or at veryhigh temperatures, then it is not true that hv>>kT and Planck'sRadiation Law must be used to model the spectral radiance from a cavityradiator.

Using the form as summarized by Dr. Einstein: (Einstein, On the QuantumTheory of Radiation, 1917)

“(Wien) discovered, as is known, the formula [Wien's radiation formula]

$\begin{matrix}{\rho = {\alpha\; v^{3}e^{\frac{- {hv}}{kT}}}} & (118)\end{matrix}$

which is recognized today as the correct limiting formula for largevalues of

$\frac{v}{T}.$

The significance of the “−1” in the denominator of Planck's Law is thatthe spectral distribution at any particular temperature will havegreater Radiance, using Planck's law, than using Wien's approximation(also sometimes called Wien's law or the Wien distribution law), at lowfrequencies (low energy). This extra Radiance is attributable to induced(stimulated) emission at low frequencies. The induced emissions do notoccur at to any discernible high frequencies (high energy). Astemperature (at which the hohlraum operates) increases, the increase inenergy is dissipated (from the hohlraum inner surface to theelectromagnetic radiation in energy equilibrium) in an increasing numberof thermal radiation photons of an average energy as opposed to everincreasing the spectral radiation energy. There is no limit to thenumber of photons that may be created in the hohlraum: by increasing thenumber of photons, N, rather than increasing the energy of each photon,the entropy

(S _(B) =−Nk _(B)Σ_(i) p _(i)ln p _(i) orS(U)=k[ln(1+U/hv)^((1+U/hv))−ln(U/hv)^(U/hv)])  (119):

of the radiation is assured to be a maximum. [In chemical reactions,entropy generally increases in reactions when the total number ofproduct molecules is greater than the total number of reactantmolecules.] This is a consequence of the Correspondence Principle—whenthe energy is large, the quantum mechanical system behaves according to(in correspondence with) the classical description. That is to say, athigh energies, the quantum mechanical representation—Planck'sLaw—corresponds to Wien's approximation and the contribution of the “−1”term in the denominator to Radiance is negligible. [Similarly, if theadjustment had been “+1” term in the denominator would reduce theunderlying energy versus the classical case and at high energies, thereduction would be minimal.]

At lower frequencies, the entropic contribution per energy at anyparticular radiation frequency is greater than the classical mechanicstreatment would otherwise project. This is a consequence of stimulatedemission in the blackbody at the lower frequencies. At frequenciesbathochromic to the entropy maximum, there is more emission of radiationfrom the walls of the hohlraum than there is absorption of the radiationby the walls. This is balanced at frequencies hypsochromic to the energymaximum where there is less emission of radiation from the walls of thehohlraum than there is absorption of radiation by the walls. In terms ofwavelength, the entropy maximum is governed by the equation:

$\begin{matrix}{{\lambda_{entropy}^{\max} = {\frac{3.00292 \times 10^{- 3}}{T}\mspace{14mu}{meters}}},{whereas}} & (120) \\{\lambda_{energy}^{\max} = {\frac{2.89777 \times 10^{- 3}}{T}\mspace{14mu}{meters}}} & (121)\end{matrix}$

such that

λ_(entropy) ^(max)/λ_(energy) ^(max)=1.03629. (Delgado-Bonal &Martin-energy Torres, 2016) (Agudelo & Cortës, 2010)  (122):

The transcendental equation for entropy maximum in wavelengths is:(Delgado-Bonal & Martin-Torres, 2016)

$\begin{matrix}{{{{- 5}\frac{hc}{\lambda\; k_{B}T}\left( \left( {e^{\frac{hc}{\lambda\; k_{B}T}} - 1} \right)^{2} \right)} - {5\frac{hc}{\lambda\; k_{B}T}\left( {e^{\frac{hc}{\lambda\; k_{B}T}} - 1} \right)} + {\frac{{hc}^{2}}{\lambda\; k_{B}T}e^{\frac{hc}{\lambda\; k_{B}T}}} - {4{\ln\left( \frac{1}{e^{\frac{hc}{\lambda\; k_{B}T}} - 1} \right)}\left( \left( {e^{\frac{hc}{\lambda\; k_{B}T}} - 1} \right)^{2} \right)} + {\left( \frac{hc}{\lambda\; k_{B}T} \right){e^{\frac{hc}{\lambda\; k_{B}T}}\left( {e^{\frac{hc}{\lambda\; k_{B}T}} - 1} \right)}}} = 0} & (123)\end{matrix}$

The expected number of photons in a photon gas is given by n=K_(n)VT³,where K_(n) is a constant. Note that for a particular temperature, theexpected particle number n varies with the volume in a fixed manner,adjusting itself to have a constant density of photons. In an idealizedsystem, should one ever exist, the change in entropy with change involume is S−S_(o)=ΔS=−nk ln (V/V_(o)). (Varró, 2006) This conceptualconstruction is essentially a two-state system, one state represented byV and the second state represented by V₀, where it is the onlyconsideration that the number of photons of thermal radiation is fixed.(Bijil, 1952) (Sokolsky & Gorlach, 2014)

When we use the term “photon” to discuss electromagnetic radiation, weare considering the energy as if it was a particle, similar to an idealgas. At high frequencies, the Wien's approximation (also sometimescalled Wien's law or the Wien distribution law) is valid (as hv>>kT):this is the region in which radiation is said to be acting like aparticle. At low frequencies, Wien's approximation is not valid, thecorrection of Planck's Radiation Law is required, and this is the regionin which the radiation is said to be acting wave-like. In reality, atleast quantum mechanical reality, the duality of radiation is simply areflection of the results of a measurement technique. The CorrespondencePrinciple articulates that the behavior of systems described by thetheory of quantum mechanics reproduces classical physics in the limit oflarge quantum numbers, high energy, and in the case of a blackbodyradiation, in the region in which the radiation is said to beparticle-light. In the region of deviation from Wien's Law, of lowquantum numbers, where the radiation from a blackbody radiator is saidto be acting wave-light, and where a correction to an otherwise validclassical mechanic formula is required (as opposed to an invalidclassical mechanic formula which yielded an obtuse description in theultraviolet).

The difference between Wien's approximation (also sometimes calledWien's law or the Wien distribution law) and Planck's Radiation Law israther small (a minus “1” in the denominator; curiously the denominatorhas a plus “1” for a Fermi oscillator) but the experimental results onblackbody radiators is of sufficient accuracy to demonstrate that thedifference is real. As Dr. Planck would no doubt relish: what more canone say? The Wien's approximation predicts the spectrum of thermalradiation for a given temperature T but deviates from the experimentalresults for long wavelengths (low frequency) emission (that is, thedeviation is apparent at lower energies of the spectral distribution).Similarly, the Rayleigh-James approximation predicts the spectrum ofthermal radiation for a given temperature T but deviates from theexperimental results for short wavelengths (high frequency) emission(that is, the deviation is apparent at higher energies of the spectraldistribution).

It has been similarly demonstrated that the fundamental differencebetween Planck's law and the Wien approximation is that the formerconsiders “induced emission” or “stimulated emission” whereas the latterdoes not. (Ross & Calvin, 1967) Stimulated emission provides for twoidentical photons each with the characteristics of the incident photon.Thus, it is called negative absorption as it appears as if lessradiation is absorbed (as one using actinometry measures more of the“incident” unabsorbed photons than is the case). The “stimulatedemission” contributions to thermal radiation emanates from or wasdetermined as a consequence of Dr. Einstein's fluctuation equation forthermal radiation.

Thus, the extra term for the second derivative of the change in entropyto energy of the oscillators and of the radiation resulted therefrom,the DU² term in

$\begin{matrix}{{- \left( \frac{d^{2}S}{{dU}^{2}} \right)^{- 1}} = {{CvU} + {DU}^{2}}} & (124)\end{matrix}$

as opposed to

$\begin{matrix}{{- \left\lbrack \frac{d^{2}S}{{dU}^{2}} \right\rbrack^{- 1}} = {C \cdot v \cdot U}} & (125)\end{matrix}$

where C is multiplied by v and by U

leads to a preference for increasing photon numbers at lowerfrequencies: thereby suggesting that there are some inefficiencies ingeneration of photons at higher frequencies. {Parenthetically, onewonders what the model would look like if the starting point was an evenlonger series; say

$\left. {\left\lbrack {{CvU} + {DU}^{2} + {\frac{F}{v}U^{3}}} \right\rbrack.} \right\}$

We recall using the nomenclature that Y₁ represents the singlet excitedstate of the secondary emitter, such as Ce³⁺:YAG, then singlet-singletannihilation can be expressed as Y₁+Y₁→Y₀+Y₂ (where Y₀ and Y2 representground electronic state and higher than the first singlet excited state,respectively) whereas stimulated emission is represented by Y₁+hv→Y₀+2hvand multi-photon absorption is represented by Y₁+hv→Y₂. Thesinglet-singlet annihilation is of the Förster Resonance Energy Transfertype and does not require the two Y₁ states to be in contact. (Andrews,2000) Indeed, stimulated emission of the type herein stated would leadto an increase in photon numbers at high frequencies as it appears to benegative absorption. Consequently, we suggest that excited statequenching of the singlet: singlet type is perhaps the reason that athigh temperatures a cavity radiator maximizes its entropy by stimulatingemission at lower frequencies and that it is excited state quenching athigher frequencies that reduces the emission to near zero.

Stimulated emission was first proposed by Dr. Einstein in 1917 byre-examining the derivation of Planck's law. “First, Einstein proposedthat an excited atom in isolation can return to a lower energy state byemitting photons, a process he dubbed spontaneous emission. Spontaneousemission sets the scale for all radiative interactions, such asabsorption and stimulated emission. Second, his theory predicted that aslight passes through a substance, it could stimulate the emission ofmore light. Einstein postulated that photons prefer to travel togetherin the same state. If one has a large collection of atoms containing agreat deal of excess energy, they will be ready to emit a photonrandomly. However, if a stray photon of the correct wavelength passesby—its presence will stimulate the atoms to release their photonsearly—and those photons will travel in the same direction with theidentical frequency and phase as the original stray photon.” (This Monthin Physics History: Einstein Predicts Stimulated Emission., 2005) Inother words, the entropy of the emission radiation field will decrease.This must be offset by an increase in entropy somewhere else.

In the case of thermal radiation, the entropy increase “somewhere else”is in the non-conservation of photon numbers. Consequently, within ablackbody cavity radiator, as the intensity, flux density, energydensity or radiance increases, the stimulated emission increases, theentropy of the radiation field decreases—because those stimulatedemission photons will travel in the same direction with the identicalfrequency and phase as the incident photons, thereby increasingorder—and the number of modes of the “emitted radiation field”increases. This is perhaps another reason for the ultravioletcatastrophe: the conservation of energy requirement of the emissionshifts from spectral energy (increasing with increasing temperatureuntil it reaches the maximum) to number of modes of radiation field(increasing with temperature). (Mungan C. E., 2005) (Mungan C., 2003)This argumentation is completely consistent with the observation thatthe entropic maximum is bathochromic to the energy maximum and that theentropy to energy ratio is less at higher frequencies, hypsochromic tothe energy maximum, and the entropy to energy ratio is greater at lowerfrequencies, bathochromic to the entropy maximum. As noted, thestimulated emission occurs at lower frequencies, the stimulated emissionis a result of greater emission than there is absorption, and thegreater emission results in an increase in the photon numbers. That isoffset by the fact that there is no stimulated emission at higherfrequencies, there is a reduction in photon numbers at frequencieshypsochromic to the energy maximum and there is greater absorption thanemissions at higher frequencies. If there was no offset as describedherein between relatively lower photon numbers at higher frequencies,and the volume of the blackbody radiator stayed the same, then theconsequence would be that the temperature of the cavity would have toincrease until a new equilibrium is obtained.

A Photon Gas. The electromagnetic radiation in a blackbody radiator,when developed using the thermodynamic and statistical mechanicsfoundations, leads one to the perhaps misleading conclusion that theradiation mirrors to some degree the thermodynamic and statisticalmechanics of an ideal gas. When integrated over all wavelengths emitted,the thermodynamic relations of state for a photon gas are well known.

In thermodynamics, the Helmholtz free energy is a thermodynamicpotential that measures the useful work obtainable from a closedthermodynamic system at a constant temperature and volume. For a photongas modelled from a black-body radiator, the Helmholtz free energy is

A=−⅓U  (126):

where A is the Helmholtz free energy and U is the internal energy of thephoton gas. Previously we noted that Helmholtz free energy isthermodynamically defined as

dA=dE−TdS.  (127):

The total internal energy of a photon gas integrated over allwavelengths or frequencies emitted from a cavity radiator is simply afunction of the constant Volume and the constant Temperature of thecavity

U=δVT ⁴  (128):

where the constant δ is a function of Boltzmann's constant k, Planck'sconstant h, and the constant speed of light c. The negative sign assuresthat the Helmholtz free energy is less than zero and this is at aminimum.

The entropy of the same photon gas, when integrated over all wavelengthsor frequencies emitted by the cavity radiator, is:

$\begin{matrix}{S = {\frac{4}{3}{\frac{U}{T}.}}} & (129)\end{matrix}$

Consequently, the term TdS=4/3 U.

So far, we have expressed the thermodynamic functions of a photon gas interms of the Internal Energy od the Photon Gas. More specifically, theinternal energy of the photon gas, U, is given by (after integratingover all frequencies): (Leff, 2002)

$\begin{matrix}{U = {\frac{8\pi^{5}k^{4}}{15c^{3}h^{3}}{VT}^{4}}} & (130)\end{matrix}$

The entropy of a photon gas is given by: (Leff, 2002)

$\begin{matrix}{S = {{\frac{4}{3}\frac{U}{T}} = {\frac{32\pi^{5}k^{4}}{45c^{3}h^{3}}{VT}^{3}}}} & (131)\end{matrix}$

At equilibrium, the Gibbs free energy is zero meaning that no moreuseful work can be obtained from the system at equilibrium. Given thethermodynamic equation

0=dG=dH−TdS;TdS=4/3U  (132):

then the enthalpy of a photon gas is: (Leff, 2002)

$\begin{matrix}{H = {{\frac{4}{3}U} = {\frac{32\pi^{5}k^{4}}{45c^{3}h^{4}}{VT}^{4}}}} & (133)\end{matrix}$

The average number of a photon gas is: (Leff, 2002)

N=rVT ³  (134):

where

$\begin{matrix}{r = {\left\lbrack {(60.4)\left( \frac{k}{hc} \right)^{3}} \right\rbrack = {a\mspace{14mu}{constant}}}} & (135)\end{matrix}$

It is emphasized that for a photon gas, the “number” is an average asopposed to an exact number whereas for an ideal gas, the “number” isexact: (PV=nRT=Nk_(B)T where k_(B)/R=Avogadro's number).

As already denoted, the Helmholtz free energy is a thermodynamicpotential used for systems at a constant temperature and a constantvolume. However, considering a cavity radiator, one cannot use the usualexpression for the chemical potential

$\left( {\frac{{dA}_{V,T}}{dN} = 0} \right)$

because one cannot increase N (i.e., add photons to the system) atconstant volume, V, and at the same time keep the temperature, T,constant. The equilibrium value for the Helmholtz free energy is notzero (recall A=−⅓ U; from TdS=4/3 U; E=U=A+TdS=−⅓ U+4/3 U=3/3 U).

The Gibbs free energy is a thermodynamic potential that measures usefulwork obtainable from a closed thermodynamic system at constanttemperature and constant pressure. At equilibrium, the Gibbs free energyis zero meaning that no more useful work can be obtained from the systemat equilibrium. The change in Gibbs free energy per particle added isknown as the chemical potential μ and since dG/dN=0 at equilibrium, thenthe chemical potential of thermal radiation from an absorbing cavityradiator is zero (0). This means that no matter how many photons arecreated or destroyed within a cavity radiator at thermal equilibrium, noadditional work is available to be performed and the number of photonscan increase and decrease and have no impact on the Gibbs free energy ofthe system. Indeed, pressure of a photon gas can be expressed solely asa function of temperature and not as a function of volume. Thus, one canadd a photon to the system at constant Temperature and keep the Pressureconstant as well.

The value of a change in Gibbs free energy is the maximal work that canproduce the system in reversible and mono-thermal conditions. Atequilibrium and at constant temperature, the Helmholtz energy functionis minimal. The difference between the Helmholtz and Gibbs energies liesin the fact that in the first case, the maximal work is equal to the sumof the useful work and that of expansion whereas in the case of theGibbs energy its decrease is only equal, in reversible conditions, tothe useful work.

The essential difference between the photon gas and the ideal gas ofmolecules: for an ideal gas, an isothermal expansion would conserve thegas energy, whereas for the photon gas, it is the energy density whichis unchanged, the number of photons is not conserved, but proportionalto volume in an isothermal change.

Optical Cooling with an Anti-Stokes' Shift. The source of primaryradiation in Down-Conversion as disclosed in U.S. Pat. No. 6,600,175 isa light emitting diode. The electroluminescence, the primary radiation,is due to electron: hole recombination, a process akin to radical ionpair recombination. In that case, an excited state of one species isformed and the ground state of the other is populated. The luminescenceis generated when the excited state is depopulated and the energy thatis available is given off in the form of electromagnetic radiation.However, Mr. Jan Tauc pointed out many years ago that a part of theenergy of electro-magnetic waves generated by electron: holerecombination (electro-luminescence produced by the passage of currentin the forward direction in a p-n junction) is taken from thesurroundings in the form of heat (entropy) and there was no barrier inprinciple to an LED being more than 100% efficient. (Tauc, 1957) Despitethis knowledge, it remains the case for light-emitting diodes, as usedwith high electrical current inputs for General Illumination,considerable heat is generated as a consequence of less than perfectelectron: hole recombination. The luminescence that is generated carriesaway both energy and entropy that is generated at the p-n junction.

In this manner, both thermal radiation and luminescence can carryentropy. Optical cooling may occur under the proper circumstances,therefore. It is well documented that electromagnetic waves from a lasercan reduce the entropy of matter; it is required however that because ofthe interaction of matter with laser radiation that the entropy ofmatter be carried away by luminescence. Since electro-magnetic wavesfrom a laser have zero entropy, and spontaneous emission fromluminescent matter so excited is spread out in all directions, this isthe most important reason why the luminescent radiation does not havezero entropy and carries away the entropy from the matter so excited.

It should not be surprising that radiation can dissipate entropy frommatter as this is what is essentially accomplished by thermal radiationemitted by a black body radiator, where the entropy carried away isshown to be a function of the temperature of the radiator:

$\begin{matrix}{S = {\frac{4}{3}\frac{U}{T}}} & (136)\end{matrix}$

where U is the internal energy of the photon gas and T is itstemperature. As noted by Mr. Mungan, the thermal radiation carries awayentropy with a 4/3 factor while the entropy of the matter decreases byU/T, and thus the total entropy change of the universe is positive inthis irreversible emission process. This implies that not only doesthermal radiation carry away entropy from matter, but it does so at arate greater than that which is lost by the matter itself. Thermalradiation has heat capacity at constant volume. One can recall that heatcapacity for matter can be determined at either constant volume orconstant pressure.

“It is possible to cool a material by anti-Stokes fluorescence. Thissimply means that the material emits photons which have a higher meanenergy than those it absorbs. The energy difference arises from thermalexcitations in the sample. Effectively, heat is converted into light,which leaves the material and is dumped onto a heat sink elsewhere.”(Mungan C., n.d.)

“The idea that anti-Stokes fluorescence might be used to cool a materialis a surprisingly old one, proposed as early as 1929 by Pringsheim.(Pringsheim, Zwei Bemerkungen Ober den Unterschied von Lumineszenz- andTemperaturstrahlung., 1929) This proposal led some 16 years later to arather spirited debate between Pringsheim and Vavilov, with the latterclaiming that its realization is impossible on thermodynamic grounds.(Vavilov, Some remarks on the Stokes law., 1945) (Pringsheim, Someremarks concerning the difference between luminescence and temperatureradiation: Anti-Stokes fluorescence., 1946) (Vavilov, Photoluminescenceand thermodynamics., 1946) Landau himself had to step into thecontroversy and proved that the entropy lost by the sample upon coolingis more than compensated for by an increase in the entropy of the light,resulting from the loss of monochromaticity, phase coherence, anddirectionality of the beam. (Landau, 1946) (Mungan C., n.d.)

“The next class of materials proposed for fluorescent cooling wassemiconductors, in a 1957 paper by a Czech theorist. (Tauc, 1957) Inthis and subsequent experimental and theoretical work until recently,the fluorescence resulted from current injection into an active junction(i.e., a LED) rather than from optical absorption using a laser.” (Keyes& Quist, 1964) (Gerthsen & Kauer, 1965) (Landsberg & Evans, 1968)(Berdahl, 1985) (Dousmanis, Mueller, Nelson, & Petzinger, 1964)(Pankove, 1975) (Mungan C., n.d.)

The mechanism by which optical cooling takes place invoking anti-Stokesfluorescence (an anti-Stokes shift) is representative of slightlyoff-resonance effects available when one uses a radiation source withnear zero entropy. The excitation source has an energy slightly belowthat which is required to populate the lowest vibrational level of anelectronically excited state. The extra energy that is required to sopopulate comes not from the radiation source but from the surroundingsin which the chromophore is located. This decreases the temperature ofthe surroundings and provides the necessary energy to populate theexcited state which then spontaneous fluoresces at an energy levelhigher than that of the excitation source. In order for an anti-Stokesshift to cool its surroundings, which is difficult to practice, theincident radiation has to be slightly off-resonant and to have almostzero entropy (which essentially means that it has to be monochromaticand of high radiance). Heretofore only a laser has been able to be apractical source of incident radiation for anti-Stokes optical cooling.

A light emitting diode such as that used in Down-Conversion for GeneralIllumination is not a laser and does not generate radiation with nearzero-entropy, especially since it is not monochromatic. It is for thisreason that the disclosure of “a microelectronic device comprising: aheat-generating structure adapted to emit a visible light output and togenerate heat energy, wherein the heat-generating structure includes atleast one of a light emitting diode and a Down-Converting luminophoricmedium; and an upconverting luminophoric material that receives aportion of the heat energy and at least partially converts the portionof the heat energy to upconverted visible light in addition to thevisible light emitted by the heat generating structure, wherein themicroelectronic device radiates the visible light output emitted by theheat-generating structure and the upconverted visible light” and“wherein the upconverting luminophoric material comprises an anti-Stokesphosphor” is not a practicable device nor a useable construction with alight emitting diode as the microelectronic device. (U.S. Pat. No.8,297,061, (2012)) The claimed or disclosed source of the incidentradiation which invokes the anti-Stokes cooling is not completely clearexcept in one description whereby upconverting phosphors absorb lightbetween 1500 nm and 1610 nm (and emits between 950 nm and 1075 nm, whichis not useful for illumination), the source of which in itself is notrevealed. Concurrently therein it states regarding FIG. 2, “Thereflecting surface 36 includes features such as reflective protrusions38 thereon that act to reflect some of the primary emission from thelight emitting diode 32 onto the anti-Stokes phosphor film 37 to enablethe cooling process” where the primary radiation is generally stated tobe visible light (in other words, light that is not 1500 nm to 1610 nm).Another inference therein of multi-photon absorption does not disclosethat such processes require very high-intensities such a laser as theexcitation source, in contrast to a light emitting diode, in general.Further complicating the claimed device is the preference that itcontain both a Stokes' phosphor and an anti-Stokes phosphor: generallyspeaking such a construction, if in a near-field implementation, leadsto quenching either by molecular collisional (near contact) interactionor by Förster resonance energy transfer (near-field virtual photonenergy transfer) so that luminescence only from the lowest energyemitter is possible. That is to say that the anti-Stokes phosphoressentially becomes a Stokes phosphor, but indirectly, provided theradiative lifetime of the anti-Stokes phosphor is substantively greaterthan 50 nano-second.

In an interesting far-field hypothetical model, whereby hohlraums withwalls containing an anti-Stokes filter, and an internal blackbody, ifone could exist, with a Stokes filter, the model shows that the internalblackbody has a lower temperature that that of the heat bath which isthe temperature at which the hohlraum operates. (McDonald, 2017) Thisrepresents an irreversible cycle and hence the model is one where thereis no equilibrium between the hohlraum and the radiation field or theinternal blackbody and the radiation field. The impracticability of sucha combined Stokes/anti-Stoke device effectively cooling at ambient isnoted. (McDonald, 2017) According to Mr. McDonald, “This result(evaporative photon cooling) is implied in . . . [J. Tauc, The Share ofThermal Energy Taken from the Surroundings in the Electro-LuminesceEnergy Radiated from a p-n Junction, Czech. J. Phys. 7, 275 (1957)] . .. as remarked by E. Yablonovitch (private communication)”. (McDonald,2017)

Thermal Properties to Non-thermal Radiation. Luminescence follows therules for Absorption and Emission as first promulgated by Dr. Einsteinwho then applied them to the equilibrium of thermal radiation (thealgorithm for which required stimulated emission as a correction toWien's Law and which incorporated wave relations for radiation as acorrection for particle relations). This may be the first timenon-thermal properties may have been applied to thermal radiation. {Oneof the more interesting examples of non-thermal properties being appliedto thermal radiation is that in its matter interacting with theradiation field, both Stokes and anti-Stokes transitions are allowed aswell as isoelectronic transitions.] But what of the converse? TheKennard-Stepanov universal relation, first developed by Mr. Kennard andthen put in modern form by Mr. Stepanov, is a thermodynamic treatment ofa transition between ground electronic states and electronic excitedstates. “The Kennard-Stepanov relation can be straightforwardly derived. . . for a system with an electronic ground state I gi and anelectronically excited state |ei, each of which are subject to anadditional sublevel structure. We assume that the excited state lifetimeis sufficiently long that its sublevel population (as well as the groundstate manifold) acquires thermal equilibrium.” The treatment is based ona derived Kennard-Stepanov function of the form:

$\begin{matrix}{{\ln\left\lbrack \frac{a(\omega)}{f(\omega)} \right\rbrack} = {\left\lbrack \frac{h\;\omega}{k_{B}T} \right\rbrack = {\left\lbrack \frac{h}{k_{B}T} \right\rbrack\lbrack\omega\rbrack}}} & (137)\end{matrix}$

In this treatment, one simply plots (for any particular wavenumber), thenatural log of the ratio of the normalized spectral intensity of theabsorption over the fluorescence (y-axis) versus the wavenumber(x-axis). This ratio then gives a slope that is

$\begin{matrix}{{{slope} = \frac{hc}{k_{B}T}};} & (138)\end{matrix}$

hence an apparent temperature can be discerned. (Moroshkin, Weller, Saß,Klaers, & Weitz, 2014)

Temperature of Radiation. We recall that the Radiance is given by:

$\begin{matrix}{{{Radiance}\mspace{14mu}{L_{v}\left( {v,T} \right)}} = {{\left\lbrack \frac{2{hv}^{3}}{c^{2}} \right\rbrack\left\lbrack \frac{1}{e^{\frac{hv}{kT}} - 1} \right\rbrack} = {{\left\lbrack \frac{2v^{2}}{c^{2}} \right\rbrack\left\lbrack \frac{hv}{e^{\frac{hv}{kT}} - 1} \right\rbrack}{W/m^{2}}{sr}}}} & (139)\end{matrix}$

The temperature can be solved in terms of the Radiance and the energy ofthe electromagnetic radiation: (Ries & McEvoy, Chemical potential andtemperature of light., 1991) (Porter, 1983) (Ries, Thermodynamics ofQuantum Conversion of Light, 1999)

$\begin{matrix}{T_{ɛ,L_{ɛ}} = \frac{ɛ}{{kln}\left\{ \frac{2ɛ^{3}}{{h^{3}c^{2}L_{ɛ}} + 1} \right\}}} & (140)\end{matrix}$

where ε=hv. The Temperature of the radiation is only dependent on theenergy (if monochromatic, there is only one energy) and the Radiance ofthe radiation. The higher the Radiance, the higher the Temperature ofthe radiation. The number of photons is given by

$\begin{matrix}{n = \frac{1}{e^{\frac{ɛ_{p} - \mu_{p}}{kT}} - 1}} & (141)\end{matrix}$

and μ is the chemical potential of the photons and ε is the energy ofthe photon. (Ries, Thermodynamics of Quantum Conversion of Light, 1999)

Recall the relationship of Spectral Radiance for blackbody thermalradiation in a vacuum:

$\begin{matrix}{{{Spectral}\mspace{14mu}{Radiance}\mspace{14mu}{L_{v}\left( {v,T} \right)}} = {{\left\lbrack \frac{2{hv}^{3}}{c^{2}} \right\rbrack\left\lbrack \frac{1}{e^{\frac{hv}{kT}} - 1} \right\rbrack} = {{{\left\lbrack \frac{2v^{2}}{c^{2}} \right\rbrack\left\lbrack \frac{hv}{e^{\frac{hv}{kT}} - 1} \right\rbrack}{Wm}} -^{2}{{Hz}^{- 1}{sr}^{- 1}}}}} & (142) \\{{Irradiance}\mspace{14mu}{or}\mspace{14mu}{Radiant}\mspace{14mu}{Flux}\mspace{14mu}{Density}\mspace{14mu}{E_{v}\left( {v,T} \right)}{\quad{\quad{= {{\left\lbrack \frac{2{\pi v}^{2}}{c^{2}} \right\rbrack\left\lbrack \frac{hv}{e^{\frac{hv}{kT}} - 1} \right\rbrack}{W/m^{2}}}}}}} & (143) \\{\mspace{25mu}{{{Radiant}\mspace{14mu}{Energy}\mspace{14mu}{Density}\mspace{14mu}{w_{e}\left( {v,T} \right)}} = {{\left\lbrack \frac{8{\pi v}^{2}}{c^{3}} \right\rbrack\left\lbrack \frac{hv}{e^{\frac{hv}{kT}} - 1} \right\rbrack}{{Joules}/m^{3}}}}} & (144)\end{matrix}$

Note that the ubiquitous term

$\begin{matrix}\left\lbrack \frac{1}{e^{\frac{hv}{kT}} - 1} \right\rbrack & (145)\end{matrix}$

Is unitless, whereas the term

$\begin{matrix}\left\lbrack \frac{hv}{e^{\frac{hv}{kT}} - 1} \right\rbrack & (146)\end{matrix}$

has units of energy. The spectral radiance has units of

Wm-² Hz⁻¹sr⁻¹

which can be expressed as

${\frac{W \cdot \sec}{m^{2} \cdot {sr}}\mspace{14mu}{as}\mspace{14mu}{Hz}^{- 1}} = {\sec.}$

This is confirmand by the units for frequency (Hz) and Planck's constant(Watts·sec²), and speed of light (m/sec).

With rearrangement, a common form for Spectral Radiance takes intoaccount that the equation is actually an integration as a function ofthe frequency:

$\begin{matrix}{{L_{v}{dv}} = {{\left\lbrack \frac{2v^{2}}{c^{2}} \right\rbrack\left\lbrack \frac{hv}{e^{\frac{hv}{kT}} - 1} \right\rbrack}{{dv}.}}} & (147)\end{matrix}$

When integrated, one obtains the Radiance and the frequency units dropsout. The formula for Radiance per unit area per steradian is obtainedafter completing the integration, whose proof has been providedelsewhere,

$\begin{matrix}{L = {{\int_{0}^{\infty}{L_{v}{dv}}} = {{\int_{0}^{\infty}{{\left\lbrack \frac{2v^{2}}{c^{2}} \right\rbrack\left\lbrack \frac{hv}{e^{\frac{hv}{kT}} - 1} \right\rbrack}{dv}}} = {\frac{2\pi^{4}k^{4}}{15h^{3}h^{3}c^{2}}T^{4}{Wm}^{- 2}{sr}^{- 1}}}}} & (148)\end{matrix}$

One can further integrate over all steradians which yields the RadianEmittance per unit area, a formula which is not necessary for theinstant invention.

An important form of the Spectral Radiance equation for thermalradiation is when one divides by the energy term hv to give the SpectralRadiance in terms of photon numbers:

$\begin{matrix}{{{Spectral}\mspace{14mu}{Radiance}\mspace{14mu}{L_{v}\left( {v,T} \right)}} = {\left\lbrack \frac{2{hv}^{3}}{{hvc}^{2}} \right\rbrack{\quad{\left\lbrack \frac{1}{e^{\frac{hv}{kT}} - 1} \right\rbrack = {{\left\lbrack \frac{2v^{2}}{c^{2}} \right\rbrack\left\lbrack \frac{1}{e^{\frac{hv}{kT}} - 1} \right\rbrack}\mspace{14mu}{photon}\mspace{14mu} s^{- 1}m^{- 2}{Hz}^{- 1}{sr}^{- 1}}}}}} & (149)\end{matrix}$

Let us consider the Spectral Radiance in terms of photon numbers byremoving the energy term hv and call this term Intensity:

$\begin{matrix}{{{Intensity}\mspace{14mu}{L_{v}\left( {v,T} \right)}} = {\frac{\left\{ {\left\lbrack \frac{2{hv}^{3}}{c^{2}} \right\rbrack\left\lbrack \frac{1}{e^{\frac{hv}{kT}} - 1} \right\rbrack} \right\}}{hv} = \left\{ {\left\lbrack \frac{2v^{2}}{c^{2}} \right\rbrack\left\lbrack \frac{1}{e^{\frac{hv}{kT}} - 1} \right\rbrack} \right\}}} & (150)\end{matrix}$

and recall that the velocity of light in a medium other than a vacuum is

$\begin{matrix}{\mspace{76mu}{{{\frac{c}{v} = n};}\mspace{76mu}{{Thus},}}} & (151) \\{{{Intensity}\mspace{14mu}{L_{v}\left( {v,T} \right)}} = {\left\{ {\left\lbrack \frac{2v^{2}}{c^{2}} \right\rbrack\left\lbrack \frac{1}{e^{\frac{hv}{kT}} - 1} \right\rbrack} \right\} = {\left\{ {\left\lbrack \frac{2n^{2}v^{2}}{c^{2}} \right\rbrack\left\lbrack \frac{1}{e^{\frac{hv}{kT}} - 1} \right\rbrack} \right\}\mspace{14mu}{photon}\mspace{14mu} s^{- 1}m^{- 2}{Hz}^{- 1}{sr}^{- 1}}}} & (152)\end{matrix}$

for Intensity of thermal radiation, in photon numbers, not in a vacuum.Generally, we do not consider the index of refraction, n, as for gasesthe value is close to unity.

To remove the units steradians⁻¹, and noting that there are 4πsteradians in a sphere, taking the Intensity, B_(v)(v, T), over 4π solidangle yields

$\begin{matrix}{{{Radiant}\mspace{14mu}{Exitance}\mspace{14mu} M_{e}} = {\left\{ {\left\lbrack \frac{8\pi\; n^{2}v^{2}}{c^{2}} \right\rbrack\left\lbrack \frac{hv}{e^{- \frac{hv}{kT}} - 1} \right\rbrack} \right\}{Wm}^{- 2}}} & (153)\end{matrix}$

This is the Spectral Radiance, now taken over 4π steradians so the unitsfor Radiant Exitance are photon Ws⁻¹ m⁻² Hz⁻¹ or Wm⁻². The RadiantExitance is the radiant flux emitted by a surface per unit area. This isthe emitted component of radiosity. “Radiant emittance” is an old termfor this quantity. This is sometimes also confusingly called“intensity”. In the form of photon numbers, the Radiant Exitance is:

$\begin{matrix}{{{Radiant}\mspace{14mu}{Exitance}\mspace{14mu} M_{e}} = {\left\{ {\left\lbrack \frac{8\pi\; n^{2}v^{2}}{c^{2}} \right\rbrack\left\lbrack \frac{1}{e^{- \frac{hv}{kT}} - 1} \right\rbrack} \right\}\mspace{14mu}{photon}\mspace{14mu} s^{- 1}m^{- 2}{Hz}^{- 1}}} & (154)\end{matrix}$

Mr. Ross and Mr. Calvin derived an equation for the change in entropywith photon numbers as: (Ross & Calvin, 1967) (Ross & Calvin, 1967)(Ross R. T., 1967)

$\begin{matrix}{{{Radiant}\mspace{14mu}{Exitance}\mspace{14mu} M_{e}} = {\left\{ {\left\lbrack \frac{8\pi\; n^{2}v^{2}}{c^{2}} \right\rbrack\left\lbrack \frac{1}{e^{- \frac{hv}{kT}} - 1} \right\rbrack} \right\}\mspace{14mu}{photon}\mspace{14mu} s^{- 1}m^{- 2}{Hz}^{- 1}}} & (154)\end{matrix}$

can be solved in terms of (where we use the notation of Ross and Calvin:

$\begin{matrix}{M_{e} = {{B\left( {v,T} \right)} = {\left\{ {\left\lbrack \frac{8\pi\; n^{2}v^{2}}{c^{2}} \right\rbrack\left\lbrack \frac{1}{e^{- \frac{hv}{kT}} - 1} \right\rbrack} \right\}\mspace{14mu}{photon}\mspace{14mu} s^{- 1}m^{- 2}{Hz}^{- 1}}}} & (156)\end{matrix}$

Generally, the term used by many in the field is brightness, signifiedby the symbol B_(v), although the term has lost favour and is encouragedto be used, if at all, in a photometric not a radiometric, context.

$\begin{matrix}{{{``{Intensity}"}\mspace{14mu}{B_{v}\left( {v,T} \right)}} = \left\{ {\left\lbrack \frac{8\pi\; n^{2}v^{2}}{c^{2}} \right\rbrack\left\lbrack \frac{1}{e^{- \frac{hv}{kT}} - 1} \right\rbrack} \right\}} & (157)\end{matrix}$

The temperature T in the equation is that of a blackbody radiator, T_(B)₀ . With some algebraic restatements, the equation can be rearranged tothe form 1/T_(B) ₀ , or hv/T_(B) ₀ :

$\begin{matrix}{{\frac{hv}{T_{B_{0}}}\left\lbrack \frac{Joules}{K} \right\rbrack} = {k_{B}\ln{\left\{ \frac{\left( {1 + {8\pi\; n^{2}v^{2}}} \right)}{c^{2}B} \right\}\left\lbrack \frac{Joules}{K} \right\rbrack}}} & (158)\end{matrix}$

where it is relied upon that we are finding for the temperature of“Intensity” of blackbody radiation expressed in terms of photon numbers(as opposed to hv, energy). (Yablonovitch, 1980) That is to say, “Theentropy gained by a blackbody upon absorption of a (“one”) photon at vis hv/T_(B) ₀ ”. (Ross R. T., 1967) One should recall that entropy inthe standard thermodynamic equation has energy over degrees K units. Oneshould also note that the temperature used, T_(B) ₀ , is simply asubstitute for radiation intensity in the form of “B_(v)(v, T)”.

Consequently, the change in entropy with the removal of one photon fromthe radiation field in terms of brightness B₁

$\begin{matrix}{{- k_{B}}\ln\left\{ \frac{\left( {1 + {8\pi\; n^{2}v^{2}}} \right)}{c^{2}B} \right\}} & (159)\end{matrix}$

and the gain in entropy due to fluorescence with the radiation field interms of brightness B₂

$\begin{matrix}{{{+ k_{B}}\ln\left\{ \frac{\left( {1 + {8\pi\; n^{2}v^{2}}} \right)}{c^{2}B} \right\}} + \frac{\left( {{hv} - u} \right)}{T}} & (160)\end{matrix}$

where T is the temperature of the surroundings.

We can further note that

$\begin{matrix}{\frac{{hv} - u}{T} = {\frac{h\left( {v_{1} - v_{2}} \right)}{T}.}} & (161)\end{matrix}$

Brightness Theorem. This theorem states that one cannot increase thebrightness of optical rays through a passive device. The term brightnessin ray optics normally is referencing “Intensity” or Radiant Exitance ashereinbefore commented. This theorem is related to entropy, reflectingon the entropy of radiation, whereby it is clear that the entropy ofbrighter, more concentrated radiation is decreased. In order to decreaseentropy, a non-spontaneous event, one has to perform work; i.e., addenergy to the system. One cannot decrease entropy without adding energyto a system, thereby eliminating a passive device to so effect.

If we consider the change in entropy with the removal of a photon from aradiation field for the purpose of absorption (the radiation can be fromany source type provided the peak frequency is v₁)

$\begin{matrix}{{{\Delta\; S_{1}} = {{- k}\;\ln\left\{ {1 + \frac{\left( {1 + {8\pi\; n^{2}v_{1}^{2}}} \right)}{c^{2}B_{1}}} \right\}}}{and}} & (162) \\{{{\Delta\; S_{2}} = {{- k}\;{\ln\left\lbrack {\left\{ {1 + \frac{\left( {1 + {8\pi\; n^{2}v_{2}^{2}}} \right)}{c^{2}B_{2}}} \right\} + \frac{h\left( {v_{1} - v_{2}} \right)}{T}} \right\rbrack}}}{and}} & (163) \\{{{\Delta\; S_{1}} + {\Delta\; S_{2}}} \geq {{0\mspace{14mu}{or}}\mspace{14mu} - {\Delta\; S_{1}} - {\Delta\; S_{2}}} \leq 0} & (164)\end{matrix}$

Then, and with some minor algebraic rearrangement

$\begin{matrix}{{{k\;\ln\left\{ \frac{\left( {1 + {8\pi\; n^{2}v_{1}^{2}}} \right)}{c^{2}B_{1}} \right\}} - {k\;\ln\left\{ \frac{\left( {1 + {8\pi\; n^{2}v_{2}^{2}}} \right)}{c^{2}B_{2}} \right\}} - \left\{ \frac{h\left( {v_{1} - v_{2}} \right)}{T} \right\}} \leq 0} & (165) \\{{{k\;\ln\left\{ \frac{\left( {1 + {8\pi\; n^{2}v_{1}^{2}}} \right)}{c^{2}B_{1}} \right\}} - {k\;\ln\left\{ \frac{\left( {1 + {8\pi\; n^{2}v_{2}^{2}}} \right)}{c^{2}B_{2}} \right\}}} \leq \left\{ \frac{h\left( {v_{1} - v_{2}} \right)}{T} \right\}} & (166) \\{{\ln\left\lbrack {\frac{\left( {1 + {8\pi\; n^{2}v_{1}^{2}}} \right)}{c^{2}B_{1}}\text{/}\frac{\left( {1 + {8\pi\; n^{2}v_{2}^{2}}} \right)}{c^{2}B_{2}}} \right\rbrack} \leq \left\{ \frac{h\left( {v_{1} - v_{2}} \right)}{T} \right\}} & (167) \\{{\frac{\left( {1 + {8\pi\; n^{2}v_{1}^{2}}} \right)}{c^{2}B_{1}}\text{/}\frac{\left( {1 + {8\pi\; n^{2}v_{2}^{2}}} \right)}{c^{2}B_{2}}} = {\left\{ {\left\lbrack \frac{v_{1}^{2}}{B_{1}} \right\rbrack{\text{/}\left\lbrack \frac{v_{2}^{2}}{B_{2}} \right\rbrack}} \right\} \leq e^{\frac{h{({v_{1} - v_{2}})}}{T}}}} & (168)\end{matrix}$

which can be rearranged to

$\begin{matrix}{\frac{B_{2}}{B_{1}} \leq {\frac{v_{2}^{2}}{v_{1}^{2}}e^{\frac{h{({v_{1} - v_{2}})}}{T}}}} & (169)\end{matrix}$

We note that the term

$\frac{v_{2}^{2}}{v_{1}^{2}}$

will always be less than one, but the term

$e^{\frac{h{({v_{1} - v_{2}})}}{T}}$

will always be greater than one and the second term will increase fasterthan the first term will decrease.

A system based on primary radiation incident on a source of secondaryradiation that results in a Stokes shift is a source of energy that mayallow for an active device to increase the brightness of the secondaryradiation (i.e.; phosphor) whereby it is seen that the greater theStokes shift, the greater the maximal brightness potential (“Intensity”or Radiant Exitance) of the secondary radiation so produced.

The temperature, T, as developed in Yablonovitch, is ambient andincludes both thermal radiation as well as radiation with a chemicalpotential. (Yablonovitch, 1980) The analogous entropy Detailed Balancedeterminations by Ross and Calvin led to brightness equations using theeffective radiation temperature of a blackbody, admittedly an artificialdesignation of temperature as alluded to by Mauzerall.

The ratio of B₂/B₁, developed by Yablonovitch for the purpose ofconcentrating light for a solar cell, shows that while the brightnessdecreases linearly as in the term

$\frac{v_{2}}{v_{1}},$

it increases exponentially via the term e^((v) ² ^(−v) ¹ ⁾ as inDown-Conversion, v₂>v₁. Whereas many have focused on Down-Conversionwith a Stokes shift engaging an obvious decrease in energy, in fact theenergy from the Stokes shift can be used to increase the brightness ofthe lower energy secondary radiation compared with the higher energyprimary radiation. The so-called energy lost from a Stokes shift, v₂−v₁,is relatively small: a 100 nm bathochromic peak shift (from 450 nm to550 nm) is equal to a loss of 11 kcal/mol. In the instant invention,another means of providing energy to increase the brightness ofsecondary radiation is provided: by using a thermal radiation source ofno more than 11 kcal/mol.

One advantage of using an infrared tertiary source to mimic the effectof Stokes shift, as in the instant invention, is one will note that atsome point, as

${v_{2}\underset{approaches}{\rightarrow}0},$

then the maximal increase in brightness B2 must approach zero, due tothe first term

$\frac{v_{2}^{2}}{v_{1}^{2}},$

whereas the second term informs that there is only thermal radiation andno radiation with chemical potential:

$e^{\frac{h{({v_{1} - v_{2}})}}{T}}$

as hv₂ approaches zero. Clearly it is better to rely on ancillarythermal radiation from the source of tertiary radiation than for theStokes shift to approach zero, which is not a practical solution in anycase for Down-Conversion, itself based on a Stokes shift for the purposeof finding the correct spectral distribution to power ANSL.

Recall that the entropy density of blackbody radiation in terms ofphoton numbers is:

S(U)=k[(1+n _(p))ln(1+n _(p))−(n _(p))ln(n _(p))]  (170):

and the change in entropy per photon number

$\begin{matrix}{\frac{dS}{{dn}_{p}} = {{k\left\lbrack {{\ln\left( {1 + n_{p}} \right)} - {\ln\left( n_{p} \right)}} \right\rbrack} = {k\left\lbrack {{\ln\left( \frac{1 + n_{p}}{n_{p}} \right)} = {{{k\left\lbrack {\ln\left( {1 + \frac{1}{n_{p}}} \right)} \right\rbrack} \cong {{- k}\;{\ln\left( n_{p} \right)}}} = \frac{hv}{T_{B_{0}}}}} \right.}}} & (171)\end{matrix}$

Radiation Transport through a Medium. A great deal of the theoreticalwork on blackbody radiators is based on transport through a vacuum. Byintroducing the refractive index, n, in the Intensity of thermalradiation, one can account for same through a transporting medium suchas a gas:

$\begin{matrix}{{{Intensity}\mspace{14mu}{B_{v}\left( {v,T} \right)}} = {\left\{ {\left\lbrack \frac{2v^{2}}{c^{2}} \right\rbrack\left\lbrack \frac{1}{e^{\frac{hv}{kT}} - 1} \right\rbrack} \right\} = \left\{ {\left\lbrack \frac{2n^{2}v^{2}}{c^{2}} \right\rbrack\left\lbrack \frac{1}{e^{\frac{hv}{kT}} - 1} \right\rbrack} \right\}}} & (172)\end{matrix}$

A gas has a standard molar entropy which is absolute. When thermalradiation is transported through a gas medium, some frequencies (energy)of the radiation are absorbed by quantum transitions, usually rotationaland vibrational and not electronic, states as well as translationalenergy. Whenever the energy of radiation is transferred to matter, thereis also a corresponding entropy transfer unless the radiation carried noentropy (coherent). The higher the standard molar entropy of the mediumabsent the radiation, the greater the entropy that can be that can betransferred from the radiation with each energy transfer (to the vapormedium) transition that is allowed. This is the principle reason thattransparent gases are preferred over transparent liquids or solids forthermal radiation entropy transfer. On the other hand, if the primaryobjective is to maintain a certain temperature and to minimize thermalradiation entropy transfer, a transparent solid (such as a glass) is apreferred embodiment.

It may be counter-intuitive that the transfer of entropy from radiationto matter is greatest when the matter itself and prior to interactionhas a high standard molar entropy. Entropy transfer is a spontaneousprocess provided the energy transfer is also a spontaneous process.Since spontaneous processes must lead to an increase in entropy, onemight wrongly conclude that the lower the entropy of the absorbingspecies, the greater the likelihood a spontaneous entropy transferprocess from radiation will satisfy the constraint that the change inentropy be positive. However, we return to the equilibrium picture ofvacuum blackbody radiation from a hohlraum whereby the equilibrium isback and forth between the subsurface of the wall matter and the thermalradiation as opposed to between the latter and the surface interfacewith the vacuum. The entropy transferred to the subsurface is fromradiation with maximal entropy for that given energy and may besubsequently transferred to a nearest-neighbour capable of oscillationbefore back to the vacuum and the creation of thermal radiation therein.If the radiation has the maximum entropy, then the oscillators withinthe walls have the maximum molar entropy. In the case of entropytransfer to gases from radiation, the higher the standard molar entropyof the vapor, the more likely a nearest-neighbour vapor can accept theentropy transfer from the vapor molecules that were initiallyentropically enhanced.

The total entropy of an ideal gas molecule can be calculated as the sumof terms, S_(Total)=S_(t)+S_(r)+S_(v)+S_(e)+S_(n)+ . . . . where thesubscripts refer to translational, rotational, vibrational, electronicand nuclear entropy terms respectively. (Kennedy, Geering, Rose, &Crossan, 2019) At standard temperature and pressure, for mostatmospheric gases, the principal contribution to the total entropy istranslational entropy whereas vibrational entropy is nearly zero.(Kennedy, Geering, Rose, & Crossan, 2019) These different elements ofstandard molar entropy are presented in Table 6 which is referred to as“Translational, rotational and vibrational entropies of gases forencapsulation in an enclosure.”

Recall that the units for entropy are Joules per degree Kelvin.

In terms of translation entropy, the vapor with the greater St, willtransfer its entropy with higher load than that with lower S_(t). Forthe following set, referred to as “Molar standard entropies of gases forencapsulation in a chamber” in Table 7, the preferred gas is that withthe highest translation entropy, calculations of which were developedfrom the Principle of Least Action. (Kennedy, Geering, Rose, & Crossan,2019)

DISCLOSURE OF THE INVENTION Technical Problem

Despite major improvements versus incandescence, inefficiencies in theproduction of illuminance from light emitting diode lamps leads to anincrease in the required electrical input to generate more radiationwhich is accompanied by a subsequent increase in the generation ofenergy that cannot be quantitatively used for work; i.e.; resulting inthe generation of heat. There is a circularity: the more useful workthat is required, the more electrical input is mandatory, the more heatthat is generated. The work for General Illumination is the utilizationof useful energy for the process of vision. One loss of efficiency isdue to the entropy of mixing, a fundamental property of Down-Conversionto form achromatic radiation that cannot be avoided.

The reduction in free energy—necessary to perform the requiredwork—associated with the entropy of mixing increases with increasingtemperature. What is necessary to improve the underlying efficiency in aprocess whereby the heat generated is dissipated in a manner that alsoincreases the amount of radiation with the spectral distributionamenable for performing useful work without the necessity for cooling.In the present invention, entropy from thermal sources includinginfra-red emitting light emitting diodes provides additional energy tovibrationally excite the ground electronic state of secondary emittersthat yields Down-Conversion.

Technical Solution

The present invention is based on the discovery that a highly efficientlight emitting device may be simply and economically fabricatedutilizing a first solid state light emitting device for generating ashorter wavelength radiation—the Primary Radiation—whose peak wavelengthis visible to a human observer. Further, at least one second lightemitting device, an emitter of thermal radiation—the infraredradiation—whose maximal peak wavelength (as defined by thetranscendental equation for entropy maximum in wavelengths, or otherwisemeasured spectroscopically) is not visible to a human observer. Further,both the Primary Radiation and the infrared radiation are transmittedto, and incident upon, a luminophore (fluorescent and/or phosphorescentmaterial, a phosphor), for Down-Conversion, by the luminophore, of theshorter wavelength radiation to yield light—Secondary Radiation. Furtherand finally, the combination of all three of which (Primary Radiation,plus Secondary Radiation, plus thermal radiation) is achromatic to ahuman observer.

There are two mechanisms for dissipation of heat by transfer: thoserelated to molecular vibrations and rotation and those independent ofintervening molecules and which can be affected in a vacuum. The formeris heat transfer by either conduction or convection, the latter is heattransfer by thermal radiation. Thermal radiation has the benefit oftransfer heat at 1⅓ time the heat content. [This “4/3” expression is aresult of equilibrium thermodynamics; “3/3(U_(r)/T) comes from the usualisovolumic entropy change of a photon gas at temperature T whenradiation of energy Ur, is added. The other U_(r)/3T comes from the workdone in expanding the volume to accommodate the new radiation at theoriginal temperature.] (Knox, Thermodynamic and the Primary Processes ofPhotosynthesis., 1969) This is an example of work being performed byheat in order to return the system back to equilibrium. Mostimportantly, in thermal radiation, there is no requirement that thenumber of photons be conserved in the transfer of heat from the source.An alternative explanation, that invokes Detailed Balance, is that theextra U_(r)/3T, compared with thermal conduction and thermal radiationtransport from emission-only, is associated with the re-absorption ofhohlraum emitted radiation so as to keep the temperature at T withoutadditional heating from the surroundings. This is a fundamentalconsequence of the Detailed Balance of matter absorbing and emittingradiation, which is not a balance for other forms of heat (thermalconduction; thermal convection). In other words, if the thermalradiation is not allowed to escape and the temperature remains at T,then the Volume of the hohlraum must increase. Alternatively, if all ofthe thermal emission is in theory allowed to transport heat to theoutside of the hohlraum, a new equilibrium must be established as thetemperature will be reduced or the temperature of the hohlraum will haveto be maintained with an external source of heat.

Down-Conversion is based on the emission of light with a chemicalpotential whereby conservation of photon numbers is required. However,as the temperature increases to a certain critical level, the chemicalpotential of secondary radiation decreases until the chemical potentialvanishes to nearly zero. Above this critical temperature, the number ofphotons of secondary radiation increases as does the number of photonsthat enhances heat transfer by thermal radiation. By eliminating heattransfer by convection and conduction and maximizing heat transfer bythermal radiation at a critical temperature at which the chemicalpotential of the secondary radiation is near zero, the photon flux ofthe light emitting diode lamp increases beyond that below the criticaltemperature.

To protect the source of primary radiation, the source of secondaryradiation and the thermal radiation (infrared radiation), any of thesesources may be isolated in an enclosure or a chamber and exposed to avacuum or a gas under a pressure that is either at, below or aboveatmospheric pressure. The gas may be used to enhance thermal migration,or it may be used to reduce thermal migration depending on how or inwhat subcomponent the device seeks to mimic a blackbody hohlraum. Thevacuum or the gas may have a protective effect, an enhancing effect orsimply transport heat from one enclosure to another enclosure. There maybe more than one enclosure and each element may be in its own enclosurewith a different gas or vacuum at different partial pressures with thepresence of air or other gases.

Advantageous Effects

There are several subcomponents of the claimed device all of which maygenerate heat or be exposed to heat as a consequence of the heatgeneration of a nearby subcomponent. Historically, General Illuminationlamps using Down-Conversion and light emitting diodes attempted totransfer all heat to the exterior of the device, to the surroundings,across the device: surroundings boundary. In the instant invention, theheat generated by the subcomponents is sought to be transported to orretained at the source of secondary radiation so as to reduce thechemical potential of the secondary radiation emitted by the phosphor,luminophore or fluorophore that is the source of secondary radiation. Atthe same time, we seek to use the gases to protect or to minimize damageof subcomponents being exposed to high temperatures. Thus, we useprotecting gases or enhancing gases such as Nobles gases and Lewis Acidor Lewis Base, respectively, whereas as well we select a smaller set ofgases that are protecting or enhancing but which have a standard molarentropy to transport to, transport away or isolate said gases to aparticular subcomponent. There are three particularly good parameters touse to select the gases that may have certain thermal properties whichcan be incorporated, depending on the design criteria. One is theStandard molar entropy of the gas; the others are the thermaldiffusivity and thermal effusivity. The latter two are a bettercriterion than the more often claimed thermal conductivity for thereasons discussed herein.

The thermal effusivity is a measure of a materials ability to exchangethermal energy with its surroundings. The thermal diffusivity of amaterial is a measure of how fast the material temperature adapts to thesurrounding temperature. Generally, they do not require much energy fromtheir surroundings to reach thermal equilibrium.

As we mentioned before, the thermal diffusivity and thermal effusivitycan be calculated with the following material properties:

(173): thermal diffusivity, a, is

$a = {\frac{\overset{¨}{\lambda}}{\rho\; C_{V}}\left\lbrack {m^{2}\text{/}s} \right\rbrack}$

(174): thermal effusivity, b, is

$b = {\sqrt[2]{\overset{¨}{\lambda}\rho\; C_{V}}\left\lbrack {{W \cdot s^{1\text{/}2}}\text{/}{m^{2} \cdot K}} \right\rbrack}$with

{umlaut over (λ)}=thermal conductivity [W/m·K]  (175):

ρ=density [kg/m³]  (176):

C _(V)=specific heat capacity [J/kg·K]  (177):

Thermal diffusivity is measured in units of metres squared per second(m² per s). Thermal diffusivity is defined to be the thermalconductivity divided by density and specific heat capacity at constantpressure. It measures the rate of transfer of heat of a material fromthe hot end to the cold end. The greater the specific heat capacity anddensity the slower the rate of transfer of heat. The thermalconductivity implies a rate of heat transfer, but it can besignificantly impacted by the capacity of the gas to hold onto the heat,thus the reason to not exclusively choose the gas based on it, thermalconductivity.

The thermal diffusivity of a material influences the penetration deptand speed of temperature adaption under a varying thermal environment.The thermal diffusivity says nothing about the energy flows. On theother hand, the thermal effusivity influences the ability to exchangethermal energy with its surroundings.

Hence, for a heat source that one wishes to transfer the heat from oneend to another, for example a light-emitting diode emitting heat and atone end of a cylindrical enclosure filled with gas whereby one seeks totransfer the heat by molecular vibrations to the other end of thecylindrical enclosure, the best parameter to use in ones' design is thehighest thermal effusivity of the otherwise transparent gas. For thesame construction but whereby one seeks to transfer thermal radiationfrom one end to another, the best parameter to use is the standard molarentropy. A table of gases and the quantity of their various “thermal”and entropy parameters is summarized in Table 8. If one seeks totransfer both heat by molecular transport and thermal radiation, it isbest to use a mixture of gases with high thermal diffusivity and highstandard molar entropy. A cylinder that contains a mixture of hydrogenand helium will find that the helium gas will separate from the hydrogengas such that the concentration gradient of helium will be in favour ofthe cold side of the cylinder: this is essentially a reversal of theentropy of mixing of distinguishable gases, such that the separationwill require a decrease in entropy, offset by the greater increase inentropy due to thermal diffusion throughout the cylinder. (Heath, Ibbs,& Wild, 1941) By way of further illustration, for a light-emitting diodeheat source at one end of a cylindrical enclosure filled with a heliumand hydrogen mixture and the cooler end having localized source ofsecondary radiation, inside the cavity the helium will have higherconcentration surrounding the phosphor and the hydrogen will have ahigher concertation surrounding the light-emitting diode.

Reported measurements of Thermal Diffusivity are strongly influenced bytemperature; the higher the temperature, the greater the reportedthermal diffusivity. The literature contains wildly variable reportednumbers if multiple sources are used as references.

BRIEF DESCRIPTION OF THE DRAWINGS

There are two drawings within the specification of the instantinvention. These drawings are labelled as “FIG. 1” and “FIG. 2”. Alldrawings incorporate therein both a light-emitting diode within orproximate to an enclosure, a secondary emitters known as phosphorswithin or proximate to the same or another enclosure, and a thermalradiation source within or proximate to the same or another enclosure.Whereas more detailed descriptions of the instant invention are providedhereinafter, we note that FIG. 1 represents a sideways view of a claimeddevice that contains the primary radiation source light-emitting diodeand adjacent thereto the thermal radiation source which may also be alight-emitting diode. This sideways view is a cross-section that cutsacross the center of the device so that all of the essential elements ofthe instant invention (primary light emitting diode, secondary source ofradiation phosphor, enclosure, a vacuum or gas, and thermal radiationdevice) are visible and, in this case, all are sealed, within theenclosure. Since the rays of light are emitted perpendicular to the topsurface in FIG. 1, this cross-section view is called a transverse view.

In a view of FIG. 1 from the top, one would not see the internalcomponents but would see a sealed rectangular enclosure, i.e.; a solidface, that appears the translucent color of the phosphor coating,internal to the enclosure. As this top-down view is not particularlyinformative nor descriptive, it is not shown in the drawings. In asimilar manner, a sideways view that is not a cross-section and viewedfrom some distance to the device sidewalls, and from which little lightis emitted as most of the light is emitted from the top face of thedevice, is not particularly informative and one would not see theinternal components but would simply see the walls of a sealedrectangular enclosure. Thus, this sideways view of the device in FIG. 1,not being informative, is not shown in the drawings.

The assembly shown in FIG. 2 is called a three-quarters view: arepresentation of the drawing posed about halfway between front andprofile views. This assembly contains and or has proximate to it theessential requirements of the claimed invention: primary light emittingdiode, secondary source of radiation phosphor, enclosure, a vacuum orgas, and thermal radiation device. The drawing in FIG. 2, because of thethree-quarters view, particularly demonstrates there are faces of theenclosure and that the enclosure is sealed such that material, matter,may not cross the boundary defined by the enclosure but that radiationand thermal radiation may so cross. FIG. 2 also shows demonstrably thatthe enclosure while sealed need not be rectangular in shape nor cubic involume. A transverse or perpendicular or parallel view of the device inFIG. 2 would not necessarily provide any additional information than thethree-quarters view and hence these other views are not presented in thedrawings of the claimed invention. Most importantly, the three-quartersview shows clearly the two faces of an enclosure, typically one throughor from which incoming primary radiation is provided, a second fromwhich scattered and or secondary radiation is delivered to the outsideenvironment, the length between these faces, the volume of space betweenthese faces and that the intervening space of a certain length need notbe uniform.

More detailed descriptions of all Figures are provided in thespecification of the instant invention but the views provided, in onecase a transverse cross-section, in another a three-quarters view,provided sufficient definition of the light-emitting diode devices tosupport that claimed herein.

BEST MODE FOR CARRYING OUT THE INVENTION

The source of primary radiation is a light-emitting diode, preferablyone comprising SiC or GaN.

The source of secondary radiation is a phosphor, otherwise called aluminophore or a luminophor.

The source of tertiary radiation is a thermal radiation source or aninfrared light-emitting diode.

The primary light-emitting diode is one that emits primary radiation andis variously described as a first solid state light emitting diode.

The secondary light-emitting diode is one that emits tertiary radiationwhich is preferably infrared radiation.

The primary radiation is emitted by the source of primary radiation.

The secondary radiation is emitted by the phosphor, luminophore orluminophor.

The tertiary radiation is non-visible radiation bathochromic to theprimary and secondary radiation.

The present invention is based on the discovery that a highly efficientlight emitting device may be simply and economically fabricated with aunique and novel construction and utilizing a first solid state lightemitting diode for generating a shorter wavelength primary radiationwhich is transmitted to, and incident upon, a luminophore (fluorescentand/or phosphorescent material) for Down-Conversion by the luminophore,of the radiation from the first solid state light emitting device, toscatter primary radiation and to separately yield secondary radiationwith a chemical potential less than that which is normally availableabsent the unique and novel construction. The chemical potential of thesecondary radiation is reduced by incorporating into the unique andnovel construction a second solid state light emitting diode forgenerating infrared radiation that is incident on the same luminophoreexposed to primary radiation such that the Down-Converted luminescencehas thermal radiation characteristics.

The general schema of an ANSL is to incorporate the following elements:

The primary radiation source is preferably light-emitting diode that isbest described as emitting primary radiation that appears to a humanobserver as being blue but which could also emit light that is perceivedto be violet or ultraviolet, but which is higher in emission energy thanthe secondary radiation source.

The secondary radiation source absorbs the primary radiation from theprimary radiation source and emits radiation, that is comprised of bothluminescence and thermal radiation, and which is called secondaryradiation.

The tertiary radiation source is one that emits radiation that is notperceived by a human observer as being visible, absent a thermal imagingdevice (e.g.; night-vision goggles), and whose radiation is incidentupon the secondary radiation source.

The primary radiation and the tertiary radiation both of which areincident upon and absorbed and or scattered by the secondary radiationsource however in the case of the tertiary radiation is principallyabsorbed by the source of secondary radiation.

The source of secondary radiation normally generates fluorescenceincluding that from triplet: triplet annihilation.

Preferably, the radiative lifetime of the secondary radiation source isless than 250 nanoseconds if the primary radiation is incoherent, andless than 100 nanoseconds if coherent and less than 25 nanoseconds iffully optically active.

Preferably, if the source of primary radiation is a semiconductor laser,the radiative lifetime of the secondary radiation source is less than 10nanoseconds.

Preferably, if the primary radiation is one that is emitted from amagnetic field, the radiative lifetime of the secondary radiation sourceis less than 70 nanoseconds.

Preferably, the source of secondary radiation is within an environmentwith a Molar Heat Capacity between 10.0 and 22.5 J mol⁻¹ K⁻¹.

Preferably, the source of primary radiation and or the source ofsecondary radiation is within an environment with S° (Standard molarentropy) of between 100 and 200 but even more preferably between 120-175J mol⁻¹ K⁻¹.

By way of example, referring to Figure No. 1 to illustrate the structureof the instant invention, a light-emitting device (101) is constructedfor emitting visible light that is perceived to be achromatic and isvisible to a human observer comprising a first light-emitting diodeelement (102) and a second light-emitting diode element (103) which havemutually different emission wavelengths and where the second element(103) emits the majority of its light not separately visible to a humanobserver and is bathochromic to the first element (102). The at leasttwo light emitting diode elements (102) and (103) emit light that isincident upon a luminescent material (104) by which radiation at awavelength bathochromic to the radiation emitted by the first lightemitting diode element (102) is produced by the luminescent material(104). In the instant invention, there is no requirement that for eachfirst light emitting diode element (102) that there be an equal amountof a second light emitting diode element (103), but there must be atleast one second light emitting diode element (103) within eachlight-emitting device (101). In one preferred embodiment, the secondlight-emitting diode element (103) emits infrared radiation at awavelength that is spectroscopically absorbed by the luminescentmaterial (104) and is at least partially transmitted through theintervening space between the second light-emitting diode element (103)and the luminescent material (4). If not spectroscopically absorbed, theluminescent material (104) at the very least must absorb the thermalradiation with an absorptivity as close to as a blackbody may be able toso do.

In another preferred embodiment, the intervening space is fullytransparent to the radiation emitted by the second light-emitting diodeelement (1033), but it may be only partially transparent in otherembodiments of the instant invention. In another preferred embodiment,the intervening space between the luminescent material (104) andneither, either or both of the light-emitting diode elements (102 and103) is in an enclosure filled with a non-cooling gas, including a gascomprised of single atoms or diatomic molecules that do not fully absorbthe spectroscopic infrared emission of the second light-emitting diode,using as a guide the publication “Infrared spectra of noble gases(12,000 to 19,000 Angstrom)” as disclosed by Mr. Humphreys and Mr.Kostkowski the United States National Bureau of Standards in the Journalof Research of the National Bureau of Standards. (Humphreys & Kostowski,2012) The gas may be a mixture of gases and each and any component mayhave a thermal effusivity and a thermal diffusivity of no less than andno more than one and twenty-five Ws^(1/2)/m²K or J/s^(1/2) m²K, and fiveand one-hundred seventy mm² per s, respectively. In another embodiment,the gas has these effusivity and diffusivity parameters as measured at300 K. In another embodiment, the gas is a mixture of no less than 98%Helium and no more than 2% hydrogen and where the mixture of the gasesis not uniform within the enclosure, when at least one of thelight-emitting diodes is operating and delivering a mixture ofradiation. Deuterium gas may replace hydrogen gas, fully or partially,wherever hydrogen gas is used.

In another preferred embodiment, the luminescent material (104) isencapsulated in an envelope or a chamber with a gas, factually not tocool, such as krypton or xenon, but otherwise to protect the luminescentmaterial (104). Reference number (105) is an insulative board having agenerally rectangular shape and made of epoxy resin and sometimesincluding silica or alumina or zeolites, with a pair of connectingelectrodes (106 a) and (106 b) for the infrared light-emitting diodeelement (3) and a pair of connecting electrodes (107 a) and (107 b) forthe blue light-emitting diode element (103). The connecting electrodes(106 a) and (106 b) for the infrared

LED element and (107 a) and (107 b) for the blue LED element arepatterned on an upper surface of the insulative board (105) and extendedto inner surfaces of through-holes (112). The chamber (118) contains aspace which comprises either a vacuum region or a gaseous region andwhich comprises (102) and (103) as previously described.

For the infrared light-emitting diode element (103), the electricalconnections to provide power to the diode is through the electrodes (106a), and through (106 b) via the wire (111). For the blue light emittingdiode element (102), the electrical connections are though theelectrodes (107 b), and (107 a) through the wire (8). The entire deviceis encapsulated through a transparent or translucent resin polymerizedto be a polymeric or encapsulating matrix beginning (109) with afull-body (thickness) shown (199) and supported via a frame (110). Theinfrared light-emitting diode element (103) may be operated in pulsemode or continuous mode, depending on the reliability of the diodeelement and the energy that must be expended to power the infraredemitting diode element (103). The part numbers of various infra-redemitting diodes available commercially or in development is presented inTable 9 and referred to as “Infrared Light Sources”.

The EVERLIGHT'S Infrared Emitting Diode (IR333-A) can be used as alow-cost infra-red light-emitting diode element (3), the forward voltageis no greater than 4 volts, and the radiant intensity per infra-redlight-emitting diode is no more than 750 mW per steradian. In thealternative to semiconductor diodes, one can use semiconductor thermalemitters such as those based on photonic crystals. (O'Regan, Wang, &Krauss, 2015)

The Down-Converting material in this embodiment comprises a yellowluminophore Ce³⁺—Nd³⁺ co-doped Y₃Al₅O₁₂ (YAG) nanoparticles, with anaverage size of 20-30 nm clusters aggregated by 8-10 nm YAGnanoparticles, and synthesized by a solvo-thermal method as describedelsewhere. (Wang, et al., 2015) When excited by blue primary radiation,strong and broad yellow luminescence (centred at 526 nm) from Ce³⁺ aswell as non-visible near-infrared (NIR) luminescence (890, 1066 and 1335nm) of Nd³⁺ is observed simultaneously. The NIR luminescence occurs viaan effective dipole-dipole energy transfer from Ce³⁺ to Nd³⁺. Bydecreasing the Nd³⁺ concentration to zero, the NIR luminescence iseliminated and the luminophore only demonstrates secondary radiationthat absent mixing with the primary radiation appears to a humanobserver to be yellow. In this embodiment, the matter that interactswith the primary radiation can be one that includes Nd³⁺ or is absentNd³⁺ but always contains Ce³⁺ and the entropy of the secondary radiationis impacted by the entropy of the incident primary radiation.

The operating scheme is as follows: 1) the primary radiation from theblue LED excites the Ce³⁺—Nd³⁺ phosphor which then emits secondaryradiation in the yellow, which when combined with the scattered blueradiation, yields achromatic radiation; 2) the infra-red emission of theNd³⁺ sites provide no benefit absent the powering of an mid-infraredLED; 3) with powering of the mid-infrared LED the Nd³⁺ sites are heatedand the emission at 890 nm is endothermically increased due to the localheating to provide red emission at energy greater than 780 nm; 4) thecombination of primary radiation, secondary radiation and theendothermic photoluminescence of the Nd³⁺ sites powered by mid-infraredLEDs provide achromatic radiation of increased yields and better colourtemperature. The radiative lifetime of the phosphor is 23 nanoseconds at0.5% Nd³⁺ loading level versus 32 nanoseconds with zero Nd³⁺

Both gallium nitride and silicon carbide LEDs are appropriate forgenerating light at appropriate wavelengths and of sufficiently highenergy and spectral overlap with absorption curves of theDown-Converting medium. The LED preferably is selected to emit mostefficiently in regions where luminescent dyes may be usefully employedto absorb wavelengths compatible with readily commercially availablefluorophores and/or phosphors for Down-Conversion to white light.

In FIG. 2, a cylindrical cavity with a radius “r” or diameter “d” andthe length, L, is as shown. The cylindrical cavity is closed on bothends with a flat face such that the interior of the cavity is able toretain a gas or a mixture of gases. On one face of the cavity is locatedproximate thereto a thermal radiation source and or light-emitting diodesources that may comprise GaN or SiC, which when powered generate light,especially radiation hypsochromic, fully or partially, to the secondaryradiation, and heat including thermal radiation. The source of primaryradiation may be a light-emitting diode, or a light-emitting diodelaser. The devices may emit coherent radiation or radiation principallyone polarization or a majority of one polarization and a minority of theother polarization in terms of handedness. These sources are shown as“210” in FIG. 2. The opposing face of the closed cylindrical cavity iscoated with phosphors “220”, either internal and in intimate contactwith the enclosed gas or mixture of gas, or external to the face. Thelength of the cylinder is coated with scattering film (230), eitherinternal and in intimate contact with the enclosed gas or mixture ofgas, or external to the long walls of the cylinder as a polymeric orencapsulating matrix (230) whose thickness is shown as (299). In thealternative, (230) may be a graphene coated covering with absorptivityof nearly 0.99. The internal space of the cylindrical cavity is enclosedwith gas or a mixture of gas; in the latter case, the mixture does nothave a uniform ratio as one gas will preferentially migrate to thecolder face and the other to the hotter face. In the case of Helium andisotopes of Hydrogen mixtures, the Helium will separate to the coldersurface and isotopes of hydrogen to the hotter surface as described in“The diffusion and thermal diffusion of hydrogen deuterium, with a noteon the thermal diffusion of hydrogen-helium”, by H. R. Heath, T. L. Ibbsand N. E. Wild. (Heath, Ibbs, & Wild, 1941)

In this way, the gas will act as a medium to transport unwanted heatfrom thermal source and light-emitting diode source to the phosphoritself. The phosphor may be kept in an environment whereby heattransferred to the phosphor so as to keep it in a hotter environmentwhen the thermal sources and light-emitting diode sources are operatedwith an electrical current, as opposed to an open circuit. The higherthe thermal diffusivity, the faster the phosphor is heated when heat,from the thermal source or light-emitting diode source, is transferred.The higher the thermal effusivity, the longer the distance “length” maybe and still engage effective transfer to heat the phosphor. Thecylinders may be mounted vertically above the light emitting diodedevice (101) such that the cylindrical face covers in a planar fashionthe entire source or sources (210) and the cylindrical face with thephosphor lies in a planar manner with the secondary radiation, thescattered radiation and any remaining infra-red radiation as emittedfrom the face as if the face was a point source.

INDUSTRIAL APPLICABILITY

The applicability of the vacuum microelectronic device with a lightemitting diode and a thermal radiation source is for the overallimprovement of General Illumination, for indoor and outdoorapplications, enhancing brightness, to the extent allowed by theconservation of energy law, despite the molecular energy lost due to theubiquitous Stokes shift employed in Down-Conversion.

1. A solid-state thermal radiation source coupled with a solid-statelight-emitting device, wherein at least one single-die semiconductorlight-emitting diode with a p-n junction is under a vacuum or exposed toa gaseous environment; wherein the at least one single-die semiconductorlight-emitting diode includes a first light-emitting diode element and asecond light-emitting diode element; wherein the second light-emittingdiode element has a spectral entropic maximum in the infrared region,comprising a GaN, InGaN, AlGaN, or AlInGaN semiconductor, or asemiconductor comprising Ga, N, In or Al configured to emit a primaryradiation which is the same for the at least one single-die LED presentin the device, said primary radiation being a relatively shorterwavelength radiation; and comprising a collection or concentrationluminophoric medium arranged in receiving relationship to said primaryradiation and tertiary radiation, wherein the luminophoric mediumresponsively emits a secondary, relatively longer wavelengthpolychromatic radiation when the luminophoric medium is excited viaexposure to the primary radiation and tertiary radiation, whereinseparate wavelengths of said polychromatic radiation mix to produce anachromatic light output; and comprising a gaseous region that comprisesa plurality of gases, whereas the at least one chamber is distinct anddifferent from a polymeric or encapsulating matrix that forms an outershape of the device.
 2. A solid-state thermal radiation source coupledwith a solid-state light-emitting device, wherein at least onesingle-die semiconductor light-emitting diode with a p-n junction isunder a vacuum or exposed to a gaseous environment, further comprising aluminescent element in a separate second enclosure and within saidsecond enclosure an environment that protects the luminescent element,wherein said environment comprises a vacuum or a partial vacuumsubstantially devoid of oxygen and wherein the second enclosure is atleast one of permanence or replaceable; wherein the at least onesingle-die semiconductor light-emitting diode includes a firstlight-emitting diode element and a second light-emitting diode element;wherein the second light-emitting diode element has a spectral entropicmaximum in the infrared, comprising a GaN, InGaN, AlGaN, or AlInGaNsemiconductor, or a semiconductor comprising Ga, N, In or Al configuredto emit a primary radiation which is the same for the at least onesingle-die LED present in the device, said primary radiation being arelatively shorter wavelength radiation; and comprising a collection orconcentration luminophoric medium arranged in receiving relationship tosaid primary radiation and tertiary radiation, wherein the luminophoricmedium responsively emits a secondary, relatively longer wavelengthpolychromatic radiation when the luminophoric medium is excited viaexposure to the primary radiation and tertiary radiation, whereinseparate wavelengths of said polychromatic radiation mix to produce anachromatic light output; and comprising a gaseous region that comprisesa plurality of gases, whereas the at least one chamber is distinct anddifferent from a polymeric or encapsulating matrix that forms an outershape of the device.
 3. The device of claim 1, wherein a gas of thegaseous region has a Standard Molar Entropy of between 100 and 200J·mol⁻¹K⁻¹.
 4. The device of claim 2, wherein a gas of the gaseousregion has a Standard Molar Entropy of between 100 and 200 J·mol⁻¹K⁻¹.5. The device of claim 1, wherein a gas of the gaseous region ishydrogen or deuterium or any mixture thereof.
 6. The device of claim 2,wherein a gas of the gaseous region is hydrogen or deuterium or anymixture thereof.
 7. The device of claim 1 wherein the infrared region isone of the Short-wave IR Type, Long-wave IR Type, or VLWIR Type.
 8. Thedevice of claim 2 wherein the infrared region is one of the Short-waveIR Type, Long wave IR Type, or VLWIR Type.
 9. The device of claim 1wherein the infrared region is the mid-infrared.
 10. The device of claim2 wherein the infrared region is the mid-infrared.
 11. The device ofclaim 1 wherein the gaseous environment is at least one of hydrogen,deuterium, ammonia, helium, oxygen, nitrogen, carbon dioxide, carbonmonoxide, argon, krypton, trifluoro-methylchloride, or xenon or anycombination thereof.
 12. The device of claim 3 wherein the infraredregion is at least one of mid-infrared, near-infrared, short-waveinfrared IR Type, long-wave IR Type, VLWIR Type.
 13. The device of claim4 wherein the infrared region is at least one of mid-infrared,near-infrared, short-wave infrared IR Type, long-wave IR Type, VLWIRType.
 14. The device of claim 11 wherein the infrared region is themid-infrared.
 15. A solid-state thermal radiation source coupled with asolid-state light-emitting device, wherein at least one single-diesemiconductor light-emitting diode with a p-n junction is under a vacuumor exposed to a gaseous environment; wherein the at least one single-diesemiconductor light-emitting diode includes a first light-emitting diodeelement and a second light-emitting diode element; wherein the secondlight-emitting diode element has a spectral entropic maximum in themid-infrared region, comprising a doped silicon carbide semiconductor,or a semiconductor comprising Ga, N, In or Al on a silicon carbidesubstrate, configured to emit a primary radiation which is the same forthe at least one single-die LED present in the device, said primaryradiation being a relatively shorter wavelength radiation; andcomprising a collection or concentration luminophoric medium arranged inreceiving relationship to said primary radiation and tertiary radiation,wherein the luminophoric medium responsively emits a secondary,relatively longer wavelength polychromatic radiation when theluminophoric medium is excited via exposure to the primary radiation andtertiary radiation, wherein separate wavelengths of said polychromaticradiation mix to produce an achromatic light output; and comprising agaseous region that comprises a plurality of gases, whereas the at leastone chamber is distinct and different from a polymeric or encapsulatingmatrix that forms an outer shape of the device.
 16. The device of claim15 wherein the gaseous environment is at least one of hydrogen,deuterium, ammonia, helium, oxygen, nitrogen, carbon dioxide, carbonmonoxide, argon, krypton, trifluoro-methyl-chloride, or xenon or anycombination thereof.
 17. The device of claim 15 wherein a gas of thegaseous region is hydrogen, deuterium or any mixture thereof.
 18. Thedevice of claim 15, wherein a gas of the gaseous region has a StandardMolar Entropy of between 100 and 200 J·mol⁻¹K⁻¹.
 19. A solid-statethermal radiation source coupled with a solid-state light-emittingdevice, wherein at least one single-die semiconductor light-emittingdiode with a p-n junction is under a vacuum or exposed to a gaseousenvironment, further comprising a luminescent element in a separatesecond enclosure and within said second enclosure an environment thatprotects the luminescent element, wherein said environment comprises avacuum or a partial vacuum substantially devoid of oxygen and whereinthe second enclosure is at least one of permanence or replaceable;wherein the at least one single-die semiconductor light-emitting diodeincludes a first light-emitting diode element and a secondlight-emitting diode element; wherein the second light-emitting diodeelement has a spectral entropic maximum in the mid-infrared, comprisinga GaN, InGaN, AlGaN, or AlInGaN semiconductor, or a semiconductorcomprising Ga, N, In or Al configured to emit a primary radiation whichis the same for the at least one single-die LED present in the device,said primary radiation being a relatively shorter wavelength radiation;and comprising a collection or concentration luminophoric mediumarranged in receiving relationship to said primary radiation andtertiary radiation, wherein the luminophoric medium responsively emits asecondary, relatively longer wavelength polychromatic radiation when theluminophoric medium is excited via exposure to the primary radiation andtertiary radiation, wherein separate wavelengths of said polychromaticradiation mix to produce an achromatic light output; and comprising agaseous region that comprises a mixture of no less than 98% helium andno more than 2% deuterium or hydrogen, whereas the at least one chamberis distinct and different from a polymeric or encapsulating matrix thatforms an outer shape of the device.
 20. The device of claim 1 wherein agas of the gaseous region has a Molar heat capacity of no less than 20.0and no more than 30.0 J·mol⁻¹ K⁻¹.